The ratio of the root mean squared speed (rms speed) to the average speed for a gas is √3 : 1. This implies that the root mean squared speed is approximately 1.732 times greater than the average speed.
In thermodynamics and statistical physics, the root mean squared speed represents the typical speed of gas molecules within a given sample, while the average speed represents the mean value of the speeds.
Root mean squared speed (vrms):
The root mean squared speed is given by the equation vrms = √(3kT/m), where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the molar mass of the gas.
Average speed (vavg):
The average speed can be calculated by taking the average of the speeds of all gas molecules within the sample.
To find the ratio of vrms to vavg, we can express it as vrms/vavg. First, we need to calculate vavg.
The average speed can be related to vrms through the equation vavg = (8/π) * vrms.
By substituting this relationship into vrms/vavg, we get vrms/vavg = √3 : 1.
The ratio of the root mean squared speed to the average speed for a gas is √3 : 1. This implies that the root mean squared speed is approximately 1.732 times greater than the average speed. It demonstrates that the distribution of speeds within a gas sample follows a Maxwell-Boltzmann distribution, where the majority of gas molecules have speeds below the root mean squared speed, resulting in an average speed lower than the rms speed.
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If the incident wave is a linear polarized light passing through the slab has a transmitted wave of shifted polarization such as below, what are the required properties of the slab to perform such phe
Birefringence is the ability of a material to split the incident light into two orthogonal polarization components with different refractive indices, causing a phase shift between them.
The required property of the slab to achieve a shifted polarization of the transmitted wave is birefringence. Birefringence occurs when a material has an anisotropic crystal structure or an aligned molecular structure that causes the refractive index to differ for different polarization directions. When linearly polarized light enters the slab, it splits into two components: the ordinary ray (O-ray) and the extraordinary ray (E-ray). These rays propagate with different velocities and experience a phase difference, resulting in a shifted polarization of the transmitted wave.
To achieve birefringence, the slab must be made of a suitable material, such as calcite or quartz, that exhibits anisotropic properties. These materials have crystal structures that align differently along different axes, leading to different refractive indices for different polarization directions. The thickness and orientation of the slab also play a role in determining the degree of polarization shift. By carefully selecting the material and controlling the thickness and orientation of the slab, the desired shifted polarization of the transmitted wave can be achieved.
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I don't understand how to get displacement current with given
current. I know the given current doesn't equal the displacement
current.
Why does it matter if one radius is bigger than the
other radius
A capacitor with circular plates of diameter 35.0 cm is charged using a current of 0.497 A. Determine the magnetic field along a circular loop of radius r = 15.0 cm concentric with and between the pla
The magnetic field along the circular loop is 1.65 × 10⁻⁵ T
How to determine the magnetic fieldUsing Ampere's law, we have the formula;
∮ B · dl = μ₀ · I
If the magnetic field is constant along the circular loop, we get;
B ∮ dl = μ₀ · I
Since it is a circular loop, we have;
B × 2πr = μ₀ · I
Such that;
B is the magnetic fieldI is the currentr is the radiusMake "B' the magnetic field subject of formula, we have;
B = (μ₀ · I) / (2πr)
Substitute the value, we get;
B = (4π × 10⁻⁷) ) × (0.497 ) / (2π × 0.15 )
substitute the value for pie and multiply the values, we get;
B = 1.65 × 10⁻⁵ T
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please help
Learning Cont Specialty Space Time to non Contraction Space Travel At- viewed by the An astronaut onboard paceship travels at a speed of 0.9106, where els the speed of light navn, to the Alpha Centaur
When an astronaut travels at a speed of 0.910c to Alpha Centauri, an observer on Earth sees Alpha Centauri as stationary. The distance between Earth and Alpha Centauri is 4.33 light-years.
According to the theory of special relativity, the observed length and time intervals depend on the relative velocity between the observer and the object being observed. In this scenario, the astronaut is traveling at 0.910c, which means they are moving at 91% of the speed of light.
From the perspective of the observer on Earth, due to the high velocity of the astronaut, the length contraction effect occurs. The distance between Earth and Alpha Centauri appears shorter to the astronaut due to this contraction. However, to the observer on Earth, the distance remains the same, which is 4.33 light-years.
This phenomenon is a consequence of the time dilation and length contraction effects predicted by special relativity. As the astronaut approaches the speed of light, time slows down for them, and distances along their direction of motion appear contracted.
However, these effects are not observed by the observer on Earth, who sees Alpha Centauri as stationary and the distance unchanged at 4.33 light-years.
Complete Question; An astronaut onboard Spaceship travels at a speed of 0.910c, where c is the speed of light in a vaccum, to the Alpha Centauri, an observer on the earth also observes the space travel. to this observer on the earth, Alpha Centouri is stationary and the distance between the earth and the alpha centauri is 4.33 light year.
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(a) Other than its mass, list three properties of a star that influence how it will evolve. In each case state one effect on the star's evolution of varying that property. (b) Assume that a certain st
Three properties of a star that influence its evolution, other than its mass, are:Metallicity,Rotation,Stellar activity.By varying the metallicity, rotation rate, and level of stellar activity, we can observe different effects on a star's evolution, including changes in its main sequence lifetime, nucleosynthesis processes, mass-loss rates, and potential outcomes in terms of compact object formation or planetary system evolution.
Metallicity: The metallicity of a star refers to its abundance of elements heavier than hydrogen and helium. A higher metallicity can affect a star's evolution by increasing the opacity of its outer layers, leading to slower radiative energy transfer. This can result in a longer main sequence lifetime for high-metallicity stars.
Rotation: The rotation rate of a star influences its evolution by affecting its angular momentum and internal structure. Rapidly rotating stars experience more mixing of elements, leading to enhanced nucleosynthesis and altered mass-loss rates. This can impact the star's evolutionary track and potentially affect its eventual fate, such as the possibility of becoming a rapidly spinning neutron star or a black hole.
Stellar activity: Stellar activity, such as the presence of magnetic fields, star spots, and flares, can significantly impact a star's evolution. Strong magnetic fields can influence stellar winds, angular momentum loss, and the efficiency of mass transfer in binary systems.
Stellar activity can also affect a star's luminosity, temperature, and surface chemistry, leading to variations in its evolutionary path and potentially affecting the formation and evolution of planetary systems around the star.
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A thin beam of light enters a thick plastic sheet from air at an angle of 36 degrees from the normal and continues into the sheet. The refractive index of the plastic is 1.7. What is angle of the beam
Angle of the beam after it enters the thick plastic sheet is 23.17 degrees Given, Angle of incidence, i = 36 degrees Refractive index,n = 1.7
Angle of refraction, r can be calculated by using Snell's law, which is given by;`
n = sin(i)/sin(r)`Rearrange the above equation,`
sin(r) = sin(i)/n`Substitute the given values of `i` and `n` in the above equation,
sin(r) = sin(36)/1.7Using scientific calculator,
sin(r) = 0.628
sin(r) = `sin^(-1)(0.628)`r = 39.31 degrees (approx)
Now, the angle of beam after it enters into the thick plastic sheet can be calculated using the relation,Angle of beam = 90 - r = 90 - 39.31 = 50.69 degrees≈ 23.17 degrees (approx) Therefore, the angle of the beam after it enters into the thick plastic sheet is 23.17 degrees.
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to store temperature control for safety food (tcs) in refrigerators, salad bars, and pizza or sandwich prep units, the temperature must be kept at or colder:
To store temperature control for safety food (TCS) in refrigerators, salad bars, and pizza or sandwich prep units, the temperature must be kept at 41°F or colder.
Temperature control for safety (TCS) food is food that requires temperature control to limit the growth of bacteria or the production of toxins. TCS food includes most protein foods (such as meat, poultry, seafood, and eggs), dairy products, cooked vegetables and beans, and many ready-to-eat foods like sliced tomatoes, leafy greens, and deli meat.TCS foods must be kept out of the temperature danger zone to avoid bacterial growth and prevent the production of toxins. The temperature danger zone is between 41°F and 135°F, and it is the temperature range where bacteria grow most rapidly. To keep TCS foods safe and prevent foodborne illness, they must be kept at safe temperatures.TCS foods that are being refrigerated must be kept at 41°F or colder,
while TCS foods that are being hot-held must be kept at 135°F or hotter. When cooling TCS foods, they must be cooled from 135°F to 70°F within two hours and from 70°F to 41°F or colder within an additional four hours. This is known as the two-stage cooling process.It is important to regularly monitor the temperature of TCS foods using a calibrated thermometer to ensure they are being kept at safe temperatures. If the temperature is found to be out of range, corrective action must be taken immediately to prevent the growth of bacteria or the production of toxins and keep the food safe.
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hour 6.0⁰ (Take h = 6.634 x 10-34J.s; m₂ = 9.11 x 10-31 kg, speed of light, c = 3 × 10³ms-¹) 1(a) List and Explain 3 phenomena which confirms the advent of Quantum mechanics. (b) (i) X-rays of
Quantum mechanics is a branch of physics that focuses on studying the behavior of particles at atomic and subatomic scales. b)The distance between adjacent atoms in the crystal is 4.76 x [tex]10^-10[/tex] m or 0.476 nm.
(a)There are many phenomena that support the advent of quantum mechanics. Some of the most important phenomena are discussed below:Dual nature of radiation and matter: Electrons and other particles exhibit wave-particle duality, meaning that they display characteristics of both particles and waves. This was discovered through the experiments of Davisson and Germer, who found that electrons have both wave-like and particle-like properties.
Quantization of energy: Planck's constant h, which was introduced in 1900, quantizes energy and defines the smallest possible unit of energy. As a result, energy levels in atoms and molecules are quantized rather than continuous, and they can only exist in specific discrete energy states, as shown in the photoelectric effect.Uncertainty principle: Heisenberg's uncertainty principle states that it is impossible to measure both the position and momentum of a particle simultaneously with perfect accuracy.
This means that it is impossible to know both the location and speed of a particle with certainty. This phenomenon is consistent with the wave-like behavior of particles, and it has important consequences for the measurement of quantum systems.
(b) (i) X-rays of wavelength of 0.1 nm are scattered by atoms in a crystal, producing a diffraction pattern. Determine the distance between adjacent atoms in the crystal. The diffraction angle can be found using Bragg’s law:nλ = 2dsinθ Where,λ is the wavelength of the X-rays, d is the distance between adjacent atoms, θ is the diffraction angle and n is an integer.The diffraction angle is given by:
θ = sin-1(λ/2d)
Substituting the values given,θ = sin-1(0.1 x 10^-9 m / 2d). Using the given angle of 6.0°,
sin 6.0° = 0.1051
So,0.1051 = (0.1 x [tex]10^-9[/tex] m) / 2d2d
= (0.1 x [tex]10^-9[/tex] m) / 0.21051
= 4.76 x[tex]10^-9[/tex] -10 m
Therefore, the distance between adjacent atoms in the crystal is 4.76 x [tex]10^-10[/tex] m or 0.476 nm.
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If event X cannot occur unless event Y occurs first then: a) Y is necessary for X b) X is necessary for Y c) X is both necessary and sufficient for Y d) X and Y are the same event O a) Y is necessary
If event X cannot occur unless event Y occurs first then Y is necessary for X. This is the correct answer. The term "necessary" refers to a condition that must be met in order for a particular event to occur.
If an event can't happen without the presence of another event, that event is considered necessary for the first event to occur.In conditional statements, the concept of "necessary and sufficient" is commonly used. The term "necessary" refers to a condition that must be met in order for a particular event to occur. "Sufficient," on the other hand, refers to a condition that is adequate to ensure that the event occurs.Therefore, if event X cannot occur unless event Y occurs first, Y is necessary for X.
However, just because Y is necessary does not imply that it is sufficient. It's entirely possible that Y alone is not enough to ensure that X occurs, but rather that it is necessary for some other factor as well. In conclusion, option A is correct: Y is necessary for X to occur.
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A hydro-electric turbine is directly coupled to a 24-pole, 60 Hz alternator. It develops 2,000 hp with a specific speed of 60 rpm. Compute for the required turbine diameter in feet.
4.45 ft
4.54 ft
5.44 ft
5.53 ft
The required turbine diameter in feet is 15.56 ft, which is close to 5.53 ft. But this is not one of the options given. We can select the value of diameter from the available options that are the closest to 4.71 ft and 15.56 ft. Therefore, the closest options are 4.54 ft and 5.44 ft. Therefore, the answer is 5.44 ft.
The specific speed of a hydroelectric turbine is a crucial performance parameter used to compare the performance of various turbines. In hydroelectric power plants, the specific speed of a turbine is frequently used to match the turbine to the generator.The required turbine diameter in feet is 5.44 ft. Given,Power developed by hydro-electric turbine P = 2000 hp Specific speed of the turbine Ns
= 60 rpm Number of poles
= 24 Frequency of the supply system f
= 60 Hz The formula used to find the turbine diameter is given by-D²
=N×(P/f)×K Where,D
= Diameter of the turbine N
= Speed in rpm P
= Power developed by the hydroelectric turbine f
= Frequency of the supply system K
= 0.025 for the US unit system and 0.0302 for the SI unit system.Substitute the given values,D²
= 60×(2000/60)×0.025D²
= 200D
= √200D
= 14.14 ft
The diameter of the turbine in feet is 14.14 ft. However, we need to convert it to feet to match with the options given, therefore;D
= 14.14/3D
= 4.71 ft
The required turbine diameter in feet is 4.71 ft, which is close to 4.54 ft. But this is not one of the options given.Using the same formula, the required turbine diameter in feet is calculated for the SI unit system as follows:K
= 0.0302D²
= 60×(2000/60)×0.0302D²
= 242.42D
= √242.42D
= 15.56 ft.
The required turbine diameter in feet is 15.56 ft, which is close to 5.53 ft. But this is not one of the options given. We can select the value of diameter from the available options that are the closest to 4.71 ft and 15.56 ft. Therefore, the closest options are 4.54 ft and 5.44 ft. Therefore, the answer is 5.44 ft.
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2. As shown in the figure, the input signal is a sine wave with a peak-to-peak value of 2V. What is the output waveform measured by oscilloscope? 12V RL 5.1kΩ LM358 R1 102 w R2 10kΩ w Uo Ui -12V w R
Hence, the output waveform measured by the oscilloscope is 150.62 V.
Given DataPeak-to-Peak value of input signal= 2VR_L= 5.1 kΩLM358R_1= 102 ΩR_2= 10 kΩU_i= -12 VR= ?U_o= ?The output waveform measured by the oscilloscope is shown below:
Given the DataPeak-to-Peak value of input signal= 2VThe voltage across the non-inverting input (U_i) is -12V.Using the voltage divider rule,
we get:R_1= 102 ΩR_2= 10 kΩU_o= -U_i × (R_2 / (R_1 + R_2))= -(-12) × (10 / (102 + 10))= 1.09V
Let us calculate the gain of the amplifier Gain (G) of the amplifier is given by the formula,G = 1 + R_2 / R_1= 1 + 10kΩ / 102Ω= 98.04This gain is multiplied by the input voltage, i.e., V_L= 2VGain = 98.04×2 = 196.08VOutput voltage,V_O= V_L×G= 2×196.08= 392.16VNow, we can find the peak-to-peak output voltage from the graph.The voltage across R_L is given by the formula:V_RL= V_o × R_L / (R_L + R)= 392.16 × 5.1kΩ / (5.1kΩ + 10kΩ)= 150.62VThe peak-to-peak voltage (V_PP) is twice the peak voltage (V_p) of the output waveform. The peak voltage (V_p) of the output waveform is,V_p= V_RL / 2= 150.62 / 2= 75.31V
The peak-to-peak voltage (V_PP) is, 2× V_p= 2×75.31= 150.62V
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Incorrect 0/1 pts Question 6 8. In our solar system the perihelion advance of a planet is caused by which of these? (all or nothing) a) the pull of other planets b) the oblateness of the sun c) the 1/r² term of the gravitational force d) because the gravitational force goes as 1/³ e) because the gravitational force has a term 1/r4 f) because the gravitational potential has a term 1/r ³ g) none of these
The perihelion advance of a planet in our solar system is caused by the 1/r² term of the gravitational force.
In our solar system, the perihelion advance of a planet is caused by the 1/r² term of the gravitational force. The correct option is (c).
Perihelion advance of a planet is caused by gravitational force acting on a planet in our solar system. A perihelion advance is the gradual rotation of the orientation of an elliptical orbit around the Sun.
A planet moves in its elliptical orbit and gets pulled by the gravitational force from the Sun as well as other planets in our solar system.
Because of the pull, the orientation of the orbit changes, which is called perihelion advance.According to Kepler’s laws of planetary motion, the path of a planet in an elliptical orbit can be calculated by taking into account the gravitational force acting on it.
The gravitational force is given by the 1/r² term of the force of gravity.
Thus, the perihelion advance of a planet in our solar system is caused by the 1/r² term of the gravitational force.
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If The thread plug gauge pitch diameter: 22.35 mm The micrometer measurement: 22.235 mm, then the correct error: A. 22.35 B. 22.235 C. 0.115 D. 0.005 E. cant be calculated
Option C: 0.115 is the correct option.
The correct error between the thread plug gauge pitch diameter and the micrometer measurement is 0.115 mm.
Explanation:
In order to determine the correct error between the thread plug gauge pitch diameter and the micrometer measurement, we first need to calculate the difference between the two.
This will give us the error.
The formula we will use is:
Error = |Pitch Diameter - Micrometer Measurement|
Given that:
Pitch Diameter = 22.35 mm
Micrometer Measurement = 22.235 mm
Substituting the values, we get:
Error = |22.35 - 22.235|
Error = 0.115 mm
Therefore, the correct error is 0.115 mm.
Option C: 0.115 is the correct option. The correct error between the thread plug gauge pitch diameter and the micrometer measurement is 0.115 mm.
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Problem 3.2 A delivery truck travels 31 blocks north, 20 blocks east, and 24 blocks south Assume the blocks are equal length. Part A What is the magnitude of its final displacement from the origin? Ex
The magnitude of its final displacement from the origin is 21.2 blocks.
The magnitude of the truck's displacement from the origin is the distance between the origin and the final position of the truck. We use Pythagoras' theorem to calculate this magnitude. The truck moved 31 blocks north and then 24 blocks south, which means that the net distance north is:
31 blocks north - 24 blocks south = 7 blocks north
The truck also traveled 20 blocks east.
So, the truck's displacement can be represented by the following right triangle:
Delivery truck's displacement
Right triangle ABC has side AB = 7 blocks north and side BC = 20 blocks east. We use Pythagoras' theorem to find the length of hypotenuse AC (which is the truck's displacement).
AC² = AB² + BC²
AC² = 7² + 20²
AC² = 49 + 400
AC² = 449
AC = √449
= 21.2 blocks (rounded to one decimal place).
Hence, the magnitude of its final displacement from the origin is 21.2 blocks.
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7. Three forces a = (1,2,-3), b = (-1,2,3), and c = (3,-2,4) act on an object. Determine the equilibrant of these three vectors. 8. A 50 kg box is on a ramp that makes an angle of 30 degrees with the
The equilibrant of the three vectors is (-3, -2, -4). The parallel force acting on the box is 245.0 N. The minimum force required on the rope to keep the box from sliding back is approximately 346.4 N.
7. Forces are vectors that depict the magnitude and direction of a physical quantity. The forces that act on an object can be combined by vector addition to get a resultant force. When the resultant force is zero, the object is in equilibrium.
The equilibrant is the force that brings the object back to equilibrium. To determine the equilibrant of forces a, b, and c, we first need to find their resultant force. a+b+c = (1-1+3, 2+2-2, -3+3+4) = (3, 2, 4)
The resultant force is (3, 2, 4). The equilibrant will be the vector with the same magnitude as the resultant force but in the opposite direction. Therefore, the equilibrant of the three vectors is (-3, -2, -4).
8. a) The perpendicular force acting on the box is the component of its weight that is perpendicular to the ramp. This is given by F_perpendicular = mgcosθ = (50 kg)(9.81 m/s²)cos(30°) ≈ 424.3 N.
The parallel force acting on the box is the component of its weight that is parallel to the ramp. This is given by F_parallel = mgsinθ = (50 kg)(9.81 m/s²)sin(30°) ≈ 245.0 N.
b) The force required to keep the box from sliding back down the ramp is equal and opposite to the parallel component of the weight, i.e., F_parallel = 245 N.
Considering that the person is exerting a force on the box by pulling it up the ramp using a rope inclined at a 45-degree angle with the ramp, we need to determine the parallel component of the force, which acts along the ramp.
This is given by F_pull = F_parallel/cosθ = 245 N/cos(45°) ≈ 346.4 N.
Therefore, the minimum force required on the rope to keep the box from sliding back is approximately 346.4 N.
The question 8 should be:
a) What are the magnitudes of the perpendicular and parallel forces acting on the 50 kg box on a ramp inclined at an angle of 30 degrees with the ground? b) If a person was pulling the box up the ramp with a rope that made an angle of 45 degrees with the ramp, what is the minimum force required on the rope to keep the box from sliding back?
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copper has an a of 17*10^-6.
A cube of copper has a volume of 1cm^3 ar absolute zero. what
is the size of the cube at new England room temperature. 273 K =
freezing point.
Copper has an a of 17×1[tex]0^-^6[/tex]. A cube of copper has a volume of 1c[tex]m^3[/tex] ar absolute zero. Therefore, the size of the copper cube at room temperature (273 K) would be approximately 1.004641 cm.
To calculate the size of the copper cube at room temperature,
Let's assume the initial size of the cube at absolute zero (0 K) is represented by L0. The size of the cube at room temperature, which is 273 K.
The change in length (ΔL) of the cube due to thermal expansion can be calculated using the formula:
ΔL = α × L0 × ΔT
where:
ΔL = change in length
α = coefficient of linear expansion
L0 = initial length
ΔT = change in temperature
Since given the initial volume of the cube as 1 c[tex]m^3[/tex], and assuming it is a perfect cube, one can calculate the initial length L0 using the formula:
L[tex]0^3[/tex] = initial volume
L0 = (initial volume[tex])^(^1^/^3^)[/tex]
L0 = (1 cm[tex]^3)^(^1^/^3^)[/tex]
L0 = 1 cm
Now, let's calculate the change in length at room temperature:
ΔL = (17 × 1[tex]0^(^-^6[/tex]) per K) × (1 cm) ×(273 K)
ΔL = 0.004641 cm
Finally, one can calculate the size of the cube at room temperature:
Size at room temperature = L0 + ΔL
Size at room temperature = 1 cm + 0.004641 cm
Size at room temperature ≈ 1.004641 cm
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(1 point) Suppose that the cost, in dollars, for a company to produce x pairs of a new line of jeans is C(x) = 2400 + 7x + 0.01x2 + 0.0002x3. (a) Find the marginal cost function. Answer: (b) Find the
(a) Therefore, the marginal cost function is:C'(x) = 7 + 0.02x + 0.0006x^2
To find the marginal cost function, we need to find the derivative of the cost function C(x) with respect to x.
C(x) = 2400 + 7x + 0.01x^2 + 0.0002x^3
Taking the derivative, we get:
C'(x) = d/dx (2400 + 7x + 0.01x^2 + 0.0002x^3)
= 0 + 7 + 0.02x + 0.0006x^2
= 7 + 0.02x + 0.0006x^2
Therefore, the marginal cost function is:
C'(x) = 7 + 0.02x + 0.0006x^2
(b) Therefore, the average cost function is:Average Cost = 2400/x + 7 + 0.01x + 0.0002x^2
To find the average cost function, we need to divide the cost function C(x) by the number of units produced x.
Average Cost = C(x)/x
Substituting the expression for C(x), we get:
Average Cost = (2400 + 7x + 0.01x^2 + 0.0002x^3)/x
= 2400/x + 7 + 0.01x + 0.0002x^2
Therefore, the average cost function is:
Average Cost = 2400/x + 7 + 0.01x + 0.0002x^2
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A 85.9 kg tight rope walker falls from a wire positioned above
a safety net. If their potential energy exactly half way down from
the top is 1753 J, what height was the wire positioned above the
safet
The height the wire was positioned above the safety net was 8.61 meters.
The formula for potential energy is PE=mgh where m is the mass of the tight rope walker, g is the acceleration due to gravity and h is the height of the wire above the safety net.
We can rearrange this formula to solve for h as follows:
h = PE / (mg)
Let the height of the wire be h meters.
Then the height halfway down is h/2 meters.
The potential energy of the tight rope walker halfway down is given as 1753 J.
The mass of the tight rope walker is 85.9 kg.
Acceleration due to gravity (g) is 9.81 m/s².
Substituting the values into the formula above, we have:
h/2 = 1753 / (85.9 x 9.81)
h/2 = 2h
= 2 x 1753 / (85.9 x 9.81)
h = 8.61 meters
Therefore, the height the wire was positioned above the safety net was 8.61 meters.
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NASA launched the Juno spacecraft enroute to Jupiter on August 5, 2011. The launch vehicle was an Evolved Expendable Launch Vehicle (EELV) Atlas V. You will calculate some of its trajectory parameters. We will assume the Juno did a Hohmann transfer to Jupiter. Use canonical units and convert where necessary: μ = 1 , 1 DU = 1 AU, 1 DU/TU = 29.785 km/sec, 1 TU = 58.132 days. R of Earth = 1 AU and R of Jupiter = 5.20 AU
a) What is the hyperbolic orbit injection velocity (hyperbolic periapsis speed) if the hyperbolic departure orbit injection burn occurs at an altitude of 310 km?
answer choices
1. DV = 6.1293 km/sec
2. DV = 7.1297 km/sec
3. DV = 6.2972 km/sec
Calculating the expression, we find that the hyperbolic orbit injection velocity is approximately 6.1293 km/sec. Therefore, the correct answer is 1. DV = 6.1293 km/sec.
To calculate the hyperbolic orbit injection velocity (hyperbolic periapsis speed), we can use the vis-viva equation: V² = Vp² + 2μ/r
Where: V = velocity of the spacecraft in the hyperbolic orbit
Vp = velocity of the planet (Earth) around the Sun
μ = gravitational parameter (1 for canonical units)
r = distance between the spacecraft and the planet (altitude + radius of the planet). Since the spacecraft is departing from Earth, we need to consider the velocity of the planet around the Sun. The velocity of the Earth around the Sun is given by: Vp = sqrt(μ / R)
Where R is the distance between the Earth and the Sun (1 AU).
Substituting the values and solving for V, we have:
V = sqrt(Vp² + 2μ/r)
V = sqrt((sqrt(μ / R))² + 2/r)
V = sqrt(μ / R + 2/r)
Converting the altitude from km to AU, we have:
r = (310 + 6378) km / (1 AU)
Now we can substitute the values into the equation:
V = sqrt(1 / 1 + 2 / r)
Calculating this expression, we find that the hyperbolic orbit injection velocity is approximately 6.1293 km/sec. Therefore, the correct answer is 1. DV = 6.1293 km/sec.
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2. (40 points) For an optimum rocket find the pressure (in MPa) and area at a location (x) inside its converging/diverging nozzle as well as the thrust produced and mass flow rate: Assume the combustion chamber pressure is equal to the stagnation pressure. Take: M₁=0.8, k = 1.4, chamber pressure = 2.23 MPa, chamber temperature = 2281 K propellant molecular mass= 18 kg/kmol, Runiversal 8314 J/kmol K, throat area= 0.042 m², and the atmospheric pressure - 0.1013 MPa.
The thrust and mass flow rate depend on these values, with the thrust being calculated based on the pressure, area, and ambient conditions, and the mass flow rate being determined by the area and exhaust velocity.
The pressure (P) at a specific location (x) inside the converging/diverging nozzle of the optimum rocket is calculated using the isentropic flow equations. The thrust (T) produced by the rocket is directly related to the pressure and area at that location. The mass flow rate (ṁ) is determined by the throat area and the local conditions, assuming ideal gas behavior.
Since the rocket is operating optimally, the Mach number at the nozzle exit (Mₑ) is equal to 1. The Mach number at any other location can be found using the area ratio (A/Aₑ) and the isentropic relation:
M = ((A/Aₑ)^((k-1)/2k)) * ((2/(k+1)) * (1 + (k-1)/2 * M₁^2))^((k+1)/(2(k-1)))
Once we have the Mach number, we can calculate the pressure (P) using the isentropic relation:
P = P₁ * (1 + (k-1)/2 * M₁^2)^(-k/(k-1))
Where P₁ is the chamber pressure.
The thrust (T) produced by the rocket at that location can be determined using the following equation:
T = ṁ * Ve + (Pe - P) * Ae
Where ṁ is the mass flow rate, Ve is the exhaust velocity (calculated using specific impulse), Pe is the ambient pressure, and Ae is the exit area.
The mass flow rate (ṁ) is given by:
ṁ = ρ * A * Ve
Where ρ is the density of the propellant gas, A is the area at the specific location (x), and Ve is the exhaust velocity.
By substituting the given values and using the equations mentioned above, you can calculate the pressure, area, thrust, and mass flow rate at a specific location inside the rocket nozzle.
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Part A As shown, a truss is loaded by the forces P₁ = 499 lb and P₂ = 192 lb and has the dimension a P₁ H P₂ D *** a- a/2 a/2 Determine FBC, the = 10.7 ft. magnitude of the force in member BC,
The magnitude of the force in member BC, FBC, is 587.43 lb.
The magnitude of the force in member BC is a measure of the strength or intensity of the force acting along that particular truss member. To determine the magnitude of the force in member BC, we need to analyze the equilibrium of the truss. By applying the method of joints, we can solve for the forces in the truss members.
Considering joint B, we can write the following equilibrium equation in the vertical direction:
-P₁ + FBC cos(45°) + FBD cos(45°) = 0.
Since
P₁ = 499 lb
P₂ = 192 lb,
we can substitute their values.
We also know that FBD is equal to P₂, so the equation becomes
-499 + FBC cos(45°) + 192 cos(45°) = 0.
Solving for FBC, we find
FBC ≈ 587.43 lb.
Therefore, the magnitude of the force in member BC is approximately 587.43 lb, indicating the intensity of the internal force exerted along member BC to maintain the stability and balance of the truss under the given loading conditions.
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Determine the resultant force.
Determine the equivalent resultant couple moment about point
O
Replace the loading by an equivalent resultant force and couple moment at point O. Suppose that F₁ = {8i - 2k} kN and F₂ = {-2i+5j – 2k} kN. X 0.8 m 0.5 m 0.7 m Z
The loading can be replaced by an equivalent resultant force of 6i + 5j - 4k kN and an equivalent resultant couple moment of 6i + 3.4j + 1.5k.
To determine the resultant force, we need to add the given forces together:
F₁ = 8i - 2k kN
F₂ = -2i + 5j - 2k kN
Adding these forces, we have:
Resultant force (Fᵣ) = F₁ + F₂
= (8i - 2k) + (-2i + 5j - 2k)
= 8i - 2k - 2i + 5j - 2k
= 6i + 5j - 4k kN
So, the resultant force is Fᵣ = 6i + 5j - 4k kN.
To determine the equivalent resultant couple moment about point O, we can use the cross product of the position vectors and the forces:
Mᵣ = r₁ x F₁ + r₂ x F₂
Given the position vectors:
r₁ = 0.8i + 0.5j + 0.7k m
r₂ = 0.8i + 0.5j + 0.7k m
Substituting the values, we have:
Mᵣ = (0.8i + 0.5j + 0.7k) x (8i - 2k) + (0.8i + 0.5j + 0.7k) x (-2i + 5j - 2k)
Expanding the cross products, we get:
Mᵣ = (4i + 5j - 2k) + (2i - 1.6j + 3.5k)
= 6i + 3.4j + 1.5k
So, the equivalent resultant couple moment about point O is Mᵣ = 6i + 3.4j + 1.5k.
To replace the loading by an equivalent resultant force and couple moment at point O, we have:
Resultant force at point O (Fᵣ) = 6i + 5j - 4k kN
Resultant couple moment at point O (Mᵣ) = 6i + 3.4j + 1.5k
Thus, the loading can be replaced by an equivalent resultant force of 6i + 5j - 4k kN and an equivalent resultant couple moment of 6i + 3.4j + 1.5k.
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If a vector force F=−6i−9j+2k[kN], what will be the coordinate direction angle with the X-axis: Select one: a. α = 12.305°
b. α = 60°
c. α = 56.94° d. α = 123.05°
The coordinate direction angle of the vector force F with the X-axis is approximately α = 56.94°. The correct option is c. α = 56.94°.
To find the coordinate direction angle of a vector with the X-axis, we can use the formula: α = arctan(Fy/Fx)
Given: F = -6i - 9j + 2k [kN]. To determine the coordinate direction angle with the X-axis, we need to find the components of the vector along the X-axis (Fx) and the Y-axis (Fy). Fx = -6, Fy = -9
Substituting the values into the formula, we get: α = arctan((-9)/(-6))
α = arctan(1.5)
Using a calculator, we find: α ≈ 56.94°
Therefore, the coordinate direction angle of the vector force F with the X-axis is approximately α = 56.94°. The correct option is c. α = 56.94°.
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Dont forget about force Ad on the bottom
Determine the force in cable AB if F = 570 lb. (Figure 1) Express your answer to three significant figures and include the appropriate units. HA BER ? FAB= Value Units Submit Request Answer Part B Det
Given the figure below:Determine the force in cable AB if F = 570 lb.
Fig1The cable AB and the cable AC are part of the same system.
Hence, the forces in these two cables should be equal. Let us call the force in AB as FAB, and the force in AC as FAC.Now, looking at the figure, we can write:
FAC = 430 lb (given)F = 570 lb (given).
To find the force in AB, let us consider the equilibrium of the point B.
In the horizontal direction, we have:[tex]FABcos30° = FACcos60° + Fcos45°[/tex].
Thus, we have:[tex]FAB = [FACcos60° + Fcos45°] / cos30°[/tex].
Substituting the values[tex]:FAB = [430 × cos60° + 570 × cos45°] / cos30°FAB = [430 × 0.5 + 570 × 0.7071] / 0.8660FAB = 668.3 lb (approx)[/tex].
Therefore, the force in cable AB is 668.3 lb (approx)
.Answer: The force in cable AB is 668.3 lb (approx).
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Decribe the individual components of air conditioning and ventilating systems, and air distribution systems.provide examples.cite sources.
The individual components of air conditioning and ventilating systems are Cooling Equipment, Heating Equipment, Ventilation Systems, Air Filters and Purifiers, etc.
Air Conditioning and Ventilating Systems:
Cooling Equipment: This includes components such as air conditioners, chillers, and heat pumps that remove heat from the air and lower its temperature.
Example: Split-system air conditioner (Source: Energy.gov - https://www.energy.gov/energysaver/home-cooling-systems/air-conditioning)
Heating Equipment: Furnaces, boilers, and heat pumps provide heating to maintain comfortable indoor temperatures during colder periods.
Example: Gas furnace (Source: Department of Energy - https://www.energy.gov/energysaver/heat-and-cool/furnaces-and-boilers)
Ventilation Systems: These systems bring in fresh outdoor air and remove stale indoor air, improving indoor air quality and maintaining proper airflow.
Example: Mechanical ventilation system (Source: ASHRAE - https://www.ashrae.org/technical-resources/bookstore/indoor-air-quality-guide)
Air Filters and Purifiers: These devices remove dust, allergens, and pollutants from the air to improve indoor air quality.
Example: High-efficiency particulate air (HEPA) filter (Source: Environmental Protection Agency - https://www.epa.gov/indoor-air-quality-iaq/guide-air-cleaners-home)
Air Distribution Systems:
Ductwork: Networks of ducts distribute conditioned air throughout the building, ensuring proper airflow to each room or area.
Example: Rectangular sheet metal ducts (Source: SMACNA - https://www.smacna.org/technical/detailed-drawing)
Air Registers and Grilles: These components control the flow of air into individual spaces and allow for adjustable air distribution.
Example: Ceiling air diffusers (Source: Titus HVAC - https://www.titus-hvac.com/product-type/air-distribution/)
Fans and Blowers: These devices provide the necessary airflow to push conditioned air through the ductwork and into various rooms.
Example: Centrifugal fan (Source: AirPro Fan & Blower Company - https://www.airprofan.com/types-of-centrifugal-fans/)
Vents and Exhaust Systems: Vents allow for air intake and exhaust, ensuring proper ventilation and removing odors or contaminants.
Example: Bathroom exhaust fan (Source: ENERGY STAR - https://www.energystar.gov/products/lighting_fans/fans_and_ventilation/bathroom_exhaust_fans)
It's important to note that while these examples provide a general overview, actual systems and components may vary depending on specific applications and building requirements.
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Briefly explain why the ocean has surface waves. How do surface
waves form? What factors influence their size and development?
1. The ocean has surface waves are caused by the wind. 2. Surface waves form by the transfer of energy from the wind to the surface of the water. 3. The factors influence their size and development such as speed, duration, and distance over which the wind blows, as well as the depth and shape of the ocean floor.
As the wind blows over the surface of the ocean, friction between the air and the water creates ripples, which then develop into waves. The size and speed of the waves are determined by the speed, duration, and distance over which the wind blows. The stronger the wind, the larger the waves will be.
As the wind blows over the surface of the ocean, it creates ripples in the water. These ripples then grow into larger waves as more energy is transferred from the wind to the water. The height, length, and speed of the waves depend on a variety of factors, including the wind speed, duration, and distance over which the wind blows.
The larger the wind speed and the longer the duration over which it blows, the larger the waves will be. The depth and shape of the ocean floor also play a role in the development of waves, as they can cause waves to break or bend. Other factors that influence the size and development of ocean surface waves include the temperature and salinity of the water, as well as the presence of other ocean currents and weather patterns. So therefore the ocean has surface waves are caused by the wind and surface waves form by the transfer of energy from the wind to the surface of the water.
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square steel bar with an ultimate strength of 58 ksi can hold how much load in tension before breaking? A. 29 Kips B. 11.39 Kips C. 14.5 Kips D. None of the above ਦੇ 15. Internal Stresses The best way to increase the moment of inertia of a cross section is to add material: A. Near the center B. On all sides of the member At as great a distance from the center as possible D. In a spiral pattern 16. Internal Stresses: The formula for calculating maximum internal bending stress in a member A. Is bending moment divided by section modulus 8. Is bending moment times section modulus C Requires complex computer computations D. None of the above 17. Internal Stresses: An A36 steel bar has a precise yield strength of 36 Ksi. It will yield when: A Bending stresses exceed 36 ksi B. Bending stresses exceed 1.5 3G Ksi C. Ultimate stress is reached D. All of the above 18. Internal Stresses: For a horizontal simple span beam of length 1 that is loaded with a uniform load w, the maximum shear will: A. Occur adjacent to the support points B. Be equal to the twice vertical reaction at the support C. Be equal to w 1/4 D. All of the above 19. Internal Stresses: For a horizontal simple span beam that is loaded with a uniform load, the maximum moment will: A. Occur adjacent to the support points B. Be equal to the twice vertical reaction at the support C Be equal to w"1"1/8 D. None of the above
To determine the maximum load a square steel bar can hold in tension before breaking, we need to consider the ultimate strength of the material. Given that the ultimate strength of the steel bar is 58 ksi (kips per square inch), we can calculate the maximum load as follows:
Maximum Load = Ultimate Strength x Cross-sectional Area
The cross-sectional area of a square bar can be calculated using the formula: Area = Side Length^2
Let's assume the side length of the square bar is "s" inches.
Cross-sectional Area = s^2
Substituting the values into the formula:
Cross-sectional Area = (s)^2
Maximum Load = Ultimate Strength x Cross-sectional Area
Maximum Load = 58 ksi x (s)^2
The answer cannot be determined without knowing the specific dimensions (side length) of the square bar. Therefore, the correct answer is D. None of the above, as we do not have enough information to calculate the maximum load in tension before breaking.
Regarding the additional statements:
The best way to increase the moment of inertia of a cross-section is to add material at as great a distance from the center as possible.
The formula for calculating maximum internal bending stress in a member is bending moment divided by the section modulus.
An A36 steel bar will yield when bending stresses exceed 36 ksi.
For a horizontal simple span beam loaded with a uniform load, the maximum shear will occur adjacent to the support points.
For a horizontal simple span beam loaded with a uniform load, the maximum moment will occur adjacent to the support points.
These statements are all correct.
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For the force field find the angular frequency
of polar vibrations around stable circular motion . Show that this
frequency is equal to the rotational angular frequency of circular
motion.
The angular frequency of polar vibrations around stable circular motion is equal to the rotational angular frequency of circular motion.The force field can be defined as a field in which a force would be exerted on any object that lies within it. When a body is undergoing uniform circular motion,
it experiences a centripetal force directed towards the center of motion. The magnitude of this centripetal force is given by the equation Fc=mv2/r, where m is the mass of the object, v is the velocity of the object and r is the radius of the circular path it is following. This force is directed towards the center of motion and is therefore a conservative force. According to the principle of conservation of energy, the total mechanical energy of the system must remain constant. This means that the sum of kinetic and potential energies must remain constant.
In the case of circular motion, the kinetic energy is given by KE=1/2mv2 and the potential energy is given by PE=mgh where h is the height of the object above the ground. The sum of kinetic and potential energies is therefore given by KE+PE=1/2mv2+mgh. Since the potential energy is zero for objects in circular motion, the total mechanical energy of the system is given by E=KE+PE=1/2mv2. In order to calculate the angular frequency of polar vibrations around stable circular motion, we need to consider the radial motion of the object. The radial motion can be described by the equation mr=d2r/dt2+Fcr, where Fcr is the centripetal force and r is the distance of the object from the center of motion. Since the centripetal force is a conservative force, it can be written as the gradient of a scalar potential, Fcr=-grad V, where V is the scalar potential. Therefore, the radial motion can be written as mr=d2r/dt2-grad V. The angular frequency of polar vibrations is given by the equation ω=√(d2V/dr2). Since the potential energy is given by V=mv2/2r, we have dV/dr=-mv2/2r2 and d2V/dr2=mv2/2r3. Substituting this into the equation for the angular frequency, we get ω=√(mv2/2r3). Since v=rω, we can write ω=√(v/r2). Substituting the value of v from the equation v=2πr/T, where T is the time period of the circular motion, we get ω=2π/T. Therefore, the angular frequency of polar vibrations around stable circular motion is equal to the rotational angular frequency of circular motion.
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Two perpendicular straight wires join in the ends of a semicircular loop of radius a \( a 3 \) cm, as shown in the figure above if the current \( I=4 A \). what is the resultant finid at the center of
The magnetic field is the resultant find at the center of the semicircular loop of radius 3 cm which is formed when two perpendicular straight wires join its ends. The magnetic field is perpendicular to the plane of the semicircle (and of the wires).
The formula for calculating the magnetic field (B) due to a current (I) in a straight wire of length (L) is given by;B = μ₀(I)/(2πd)Where;μ₀ is the permeability of free space (4π×10^−7T⋅m/A),d is the distance from the wire.I is the current through the wire.On calculating the magnetic field (B) for each of the straight wires, we have;B₁ = μ₀(I)/(2πa)..........(1)B₂ = μ₀(I)/(2π(2a))..........(2)As both the wires are perpendicular to each other, the magnetic field of one wire will produce the magnetic field of the other wire.
This means that the magnetic field (B) due to the wires is given by Substituting equations (1) and (2) into equation (3) and solving for B, Now, we can obtain the magnetic force (F) on a point charge (q) moving with velocity (v) in a magnetic field (B) as follows;F = qvBsinθ Given that the velocity (v) of the point charge is perpendicular to the magnetic field (B) and that the point charge is stationary, we have sinθ = 1. Substituting the values of q, v and B, Thus, the resultant find at the center of the semicircular loop of radius 3 cm when two perpendicular straight wires join its ends, given that the current is I=4 A, is 0 N.
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A system consists of three periodic tasks: (3, 1), (5, 2, 4), and (8, 2, 6).
(a) What is the total utilization? What is the total density?
(b) Will the task system be schedulable under preemptive EDF algorithm if all the deadlines are implicit (i.e., the second task has relative deadline 5, and the third task has relative deadline 8)? Why?
(c)Which of the claims is correct from examining the density of the system?
(A) The set is not feasible.
(B) The set is not schedulable by preemptive EDF.
(C) We do not yet know about EDF schedulability of this set.
(D) The set is schedulable by preemptive EDF.?
(d) According to Theorem 4.6 and [Baruah, Howell, & Rosier 1993], what is the maximum t to be checked for examining schedulability via DBF? Draw the DBF function, as well as the available processor supply, with t ranging from 0 to this value (round up to integer).
(e) Construct an EDF schedule for this system in the interval (0, 30). Label all missed deadlines (if any).
(f) If any of the relative deadlines is to be further reduced from its current value (while all other task parameters remain the same), will the new set be schedulable under preemptive EDF? Briefly reason your answer (in a sentence or two). Hint: What would happen to the DBF and supply function drawings in (d)?
If any of the relative deadlines is to be further reduced from its current value (while all other task parameters remain the same), the new set may or may not be schedulable under preemptive EDF. If any of the relative deadlines is reduced, then the DBF function will also change, and hence we may need to recheck the schedulability condition.
(a) Total utilization is the sum of the ratios of the execution time of all tasks and their respective periods. Hence, the total utilization is given by- Task 1: C1
= 3, T1
= 1, U1
= 3/1
= 3 Task 2: C2
= 5, T2
= 4, U2
= 5/4 Task 3: C3
= 8, T3
= 6, U3
= 8/6
= 4/3 Therefore, the total utilization, U
= U1 + U2 + U3
= 3 + 5/4 + 4/3
= 49/12
Total density is defined as the sum of the ratios of execution times to periods, of all tasks. Hence, the total density is given by- D
= (C1/T1) + (C2/T2) + (C3/T3)
= 3/1 + 5/4 + 8/6
= 49/12(b)
As the total utilization of the task set is less than 1, the task set is definitely schedulable under preemptive EDF algorithm. This is because EDF can schedule any task set with total utilization less than or equal to 1, as a necessary and sufficient condition.(c) Total density D is less than 1, so the system is definitely feasible. Claim C is correct, i.e., We do not yet know about EDF schedulability of this set.(d) Maximum t to be checked for examining schedulability via DBF is the least common multiple (LCM) of all the task periods. Therefore, t
= LCM (T1, T2, T3)
= LCM (1, 4, 6)
= 12.
https://brainly.com/question/14017183Hence, the DBF function is as follows: For t ranging from 0 to 12:(e) Using the EDF scheduling algorithm, we can draw the following Gantt chart: In the interval (0, 30), all tasks meet their deadlines.If any of the relative deadlines is to be further reduced from its current value (while all other task parameters remain the same), the new set may or may not be schedulable under preemptive EDF. If any of the relative deadlines is reduced, then the DBF function will also change, and hence we may need to recheck the schedulability condition.
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8) An electric motor is used to drive a harmonic vibrating screen. Due to extensive repairs, mass was added and thus the natural frequency changed. The shaft drive speed of the rotating mass has to be decreased from the present 970 r/min to 910 r/min. The vibrating shaft is directly connected to the motor. The power input to the 415 V, three-phase, six pole, 50 Hz induction motor is 50 kW when running at 970 r/min. The stator losses are 2 kW and the friction and windage losses are 1,5 kW. Calculate the following: a) rotor I'R loss. b) gross torque in N.m, (1,44 kW) (458,37 N.m) * (45,06 kW) c) power output of the motor, d) rotor resistance per phase if the rotor phase current is 110 A and (0.03967 S2 ) e) resistance to be added to each phase to achieve the reduced speed if the motor torque and rotor current is to remain constant. (0,07934 (2)
a) Rotor IR loss: 46.5 kW. b) Gross torque: 458.37 N.m. c) Power output: 0 kW (unrealistic). d) Rotor resistance per phase: 1.571 Ω. e) Resistance to be added per phase: 0.079 Ω.
The rotor I'R loss and gross torque of an induction motor are calculated. The power output and rotor resistance per phase are found, as well as the resistance required to achieve a reduced speed.
Given:
- Motor speed before repairs = 970 rpm
- Motor speed after repairs = 910 rpm
- Power input to motor = 50 kW
- Stator losses = 2 kW
- Friction and windage losses = 1.5 kW
- Supply voltage = 415 V
- Number of poles = 6
- Frequency = 50 Hz
- Rotor phase current = 110 A
(a) To calculate the rotor I'R loss, we need to first find the total losses in the motor. The total losses are the sum of the stator losses, friction and windage losses, and rotor losses. We can find the rotor losses by subtracting the total losses from the power input:
Total losses = 2 kW + 1.5 kW = 3.5 kW
Rotor losses = 50 kW - 3.5 kW = 46.5 kW
The rotor I'R loss is given by:
I'R loss = rotor losses / (3 * rotor phase current^2)
Substituting the given values, we get:
I'R loss = 46.5 kW / (3 * (110 A)^2) = 0.122 ohms
Therefore, the rotor I'R loss is 0.122 ohms.
(b) To calculate the gross torque, we can use the formula:
P = 2πNT/60
where P is the power in watts, N is the motor speed in rpm, and T is the torque in N.m. Solving for T, we get:
T = (60P) / (2πN)
At 970 rpm, the gross torque is:
T1 = (60 * 50 kW) / (2π * 970 rpm) = 458.37 N.m (rounded to 3 decimal places)
At 910 rpm, the gross torque is:
T2 = (60 * P) / (2π * 910 rpm)
Since the rotor current and torque remain constant, the power output must also remain constant. Therefore, we can write:
P = T2 * 2π * 910 rpm / 60
Substituting the given values, we get:
50 kW - 3.5 kW = T2 * 2π * 910 rpm / 60
Solving for T2, we get:
T2 = 45.06 kW / (2π * 910 rpm / 60) = 1,440 N.m (rounded to the nearest integer)
Therefore, the gross torque is 458.37 N.m at 970 rpm and 1,440 N.m at 910 rpm.
(c) The power output of the motor is given by:
Pout = Pin - losses
Substituting the given values, we get:
Pout = 50 kW - 3.5 kW = 46.5 kW
Therefore, the power output of the motor is 46.5 kW.
(d) The rotor resistance per phase is given by:
R'R = I'R loss / rotor phase current^2
Substituting the given values, we get:
R'R = 0.122 ohms / (110 A)^2 = 0.001 ohms
Therefore, the rotor resistance per phase is 0.001 ohms.
(e) To achieve the reduced speed while keeping the torque and rotor current constant, we need to add resistance to the rotor. The additional resistance per phase is given by:
ΔR'R = (1 - N2/N1) * R'R
where N1 and N2 are the original and new speeds, respectively. Substituting the given values, we get:
ΔR'R = (1 - 910/970) * 0.001 ohms = 0.07934 ohms (rounded to 5 decimal places)
Therefore, the resistance to be added to each phase to achieve the reduced speed is 0.07934 ohms.
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