The length of rotating string having 47 revolutions is 2m.
A string trimmer is a tool for cutting grass and weeds; it makes use of a length of nylon "string" that rotates around an axis perpendicular to one end of the string. The string's tangential speed at its tip is 54 m/s, and its angular speed is 47 rev/s.
[tex]Z=dc+CvdT+\pi (d^2L)/dT^2+\alpha \\\\Z=dc+CvdT\\\\Z=\left \{ {{\neq 1} \atop {=1}} \right.[/tex]
For ideal situations Z needs to be unity.
Mathematically, function tends to reduce to negligible.
This is due to its inverse relation with order of the derivative.
[tex]Cv\geq 1[/tex]
The linear speed of any object moving along a circular path at any given moment is known as tangential speed or tangential velocity. For any two instants of time for a given object in circular motion, the direction of tangential speed can never be the same.
[tex]\frac{d^2L}{y}+S=0\\ \\\frac{d^2L}{y}=0\\\\s\neq 0[/tex]
[tex]speed= \frac{47\pi }{54}\\ \\=12.38m/s[/tex]
Length of string=12.38 /2π
=1.97m approx to 2m
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Cations are always _____ than the parent atom and anions are always _____ than the parent atom.
Cations are always smaller than the parent atom and anions are always larger than the parent atom.
What are cations and anions?Cations are positively charged ions i.e. one that would be attracted to the cathode in electrolysis while anions are negatively charged ions.
When a balanced atom loses one or more electrons, it will become a positively charged cation but it becomes negatively charged when it accepts one or more electrons.
Cations are always smaller than their parent atoms because they have lesser electrons, while their nuclear charge remains the same. On the other hand, anions are always larger because they have more electrons than their parent atoms.
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a satellite is orbiting a planet a distance r from its center and another satellite is orbiting at a distance 3r from its center. what is the relation between the accelerations due to gravity for each case?
Answer:
The relation between acceleration due to gravity for both cases [tex]g_{1} = 9g_{2}[/tex]
Explanation:
What is acceleration due to gravity ?
Any object's acceleration when it is only being affected by gravity.
According to the given information
Acceleration due to gravity could be found by [tex]\frac{GM}{r^{2} }[/tex]
Distance between satellite orbiting planet from center = r
Acceleration due to gravity for this case [tex]g_{1}[/tex] = [tex]\frac{GM}{r^{2} }[/tex]
Distance between satellite orbiting planet from center = 3r
Acceleration due to gravity for this case [tex]g_{2}[/tex] = [tex]\frac{GM}{(3r)^{2} }[/tex]
relation between the acceleration due gravity in each case
[tex]\frac{g_{1} }{g_{2} } =\frac{(3r)^{2} }{r^{2} } \\[/tex]
[tex]\frac{g_{1} }{g_{2} } =\frac{9r^{2} }{r^{2} } \\[/tex]
[tex]\frac{g_{1} }{g_{2} }[/tex] = 9
[tex]g_{1} = 9g_{2}[/tex]
The relation between acceleration due to gravity for both cases [tex]g_{1} = 9g_{2}[/tex]
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Show that the Poisson bracket of two constants of the motion is itself a constant of the motion, even when the constants depend upon time explicitly.
Poisson brackets are powerful and sophisticated tool in the Hamilton formalism of classical mechanism.
The nonverbal code that most explicitly deals with movements and motion-based behaviors is known as kinesics.
Kinesics in Poisson bracket describes the way our hands, arms, faces, and bodies move to express meaning. It's our body language throughout a discussion or how we appear when we're speaking. This might show that we are interested in the interaction and are confident in it.
Poisson bracket refers to a broad range of visual codes involving the movement of the human body. Among these codes are gestures, posture, eye gaze, and facial expressions. Artifacts represent the various ways that we adorn or manipulate the body to communicate cultural or personal meanings.
Therefore, these Poisson bracket also shows a link between quantum and classical mechanism.
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suppose that a planet was reported to be orbiting the sun-like star iota horologii with a period of 300 days. find the radius of the planet's orbit, assuming that iota horologii has four times the mass as the sun.
Radius of the planet's orbit is 2.15 * [tex]10^{-6}[/tex] m
Radius of the orbit can be calculated by applying Kepler's third law, which compares the motion characteristics of different planets, i.e. it compares the orbital period and the radius of the orbit.
Mathematically, [tex]T^{2} = 4\pi ^{2} * r^{3} / GM[/tex]Where
T = Time period = 300Days ( Period)
r = radius
G = Gravitational constant = 6.67 * [tex]10^{-11}[/tex] m³/Kgs
M = Mass of the object, sun in this case = 1.99 * [tex]10^{30}[/tex] kg
Then, rearranging the Kepler's equation,
[tex]R^{3} = GMT^{2} /4\pi ^{2}[/tex]
R³ = 100 * [tex]10^{-19}[/tex]
R = 2.15 * [tex]10^{-6}[/tex] m
The radius of the planet which is reported to orbiting for 300 days is 2.15 * [tex]10^{-6}[/tex] m
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A lab technician uses laser light with a wavelength of 640 nm to test a diffraction grating. When the grating is 40.8 cm from the screen, the first-order maxima appear 6.00 cm from the center of the pattern.
A lab technician uses laser light with a wavelength of 640 nm to evaluate a diffraction grating, and this particular grating has 195.31 lines per millimeter.
Its wavelength is 640 nm.
L=40.8 cm The distance between the grating and the screen
The first dazzling fringe is six centimeters away from the centre of the order for laser light. According to the deduced results from Young's experiment, which tie the order of bright fringes to the central order,
With the approach of laser light is small-angle approximation, y = m(wavelength)(L)/d and d = m()(L)/y
Using the value d= 1*640*10^-9*0.48/0.06m = 5.120*10^-9 as a starting point
The line density is 1/d = 1/5.120*10^-9, or 195.31 lines per millimeter.
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escribe the following metrology tools: scales, calipers, coordinate measuring machine, and 3-d laser scanner. which is least expensive? which is most accurate?
3D lasers scanner: uses laser projector and cameras to determine part dimensional coordinates to high degree of accuracy, high cost.
Scales: similar to a ruler used for linear measures with "eyeball" accuracy, traditional, affordable, and occasionally insufficiently exact.
Measurements are fairly accurate thanks to the caliper's ability to expand to fit the part being measured.
Coordinate measuring machine: uses probe to touch part and senses x, y, and z coordinate locations that record information into memory (also known as digitizing model), accuracy in the 0.001 mm range, larger volumes greatly increase system cost
No of the size or surface details of the object being scanned, 3D laser scanner uses either phase-based or LIDAR technology to reliably and effectively gather 3-dimensional data. 3D laser scanning generates clear and accurate digital records of present conditions without actually touching what is being measured. The scan generates countless numbers of measurement points known as "coordinates." "Point clouds" are collections of points that together represent raw data. An x, y, and z value is contained in each coordinate in a point cloud. These values are often referred to as northing (y), easting (x), and elevation (z). These points have such fine precision that it is possible to measure exactly from one point to any other point in the point cloud.
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what is the potential energy u of the toy when the spring is compressed 3.2 cm from its equilibrium position?
Total energy in compressing a spring is [tex]kx^{2}/2[/tex].
When we compress or extend a stretched spring, we experience a force equal to that applied by us in the opposite direction.When a spring deviates from its mean position, it tends to restore its equilibrium by exerting a force equal and opposite to the external force. We have seen the application of spring force in bicycle carriers and launching devices where the energy gained by disturbing the equilibrium of the spring is utilized as its potential energy and converted to other forms.To know more about springs visit:
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A book weighing 10 kg is preparing to fall off the desk from a 1 meter height. The book's
gravitational potential energy is 980 Joules.
O True
O False
Answer:
False
Explanation:
gravitational potential energy = mgh
10*10*1 = 100 joules
what magnitude of current is required to produce 1.4 kg of sodium metal in one hour? express your answer in amperes to two significant figures.
The magnitude of the current required to produce 1.4 kg of sodium metal in one hour is 1398.38 Amps
Weight of Sodium = 1.4 kg = 1400 g
The molar mass of Sodium = 23 u
Moles of Sodium = 1400/23 = 52.174
No. of atoms of copper = 52.174 * 6.023 x[tex]10^{23}[/tex] = 3.14 x [tex]10^{25}[/tex]
Each copper requires a 1e charge,
So total charge = 1.602 x[tex]10^{-19}[/tex] * 3.14 x [tex]10^{25}[/tex]= 5034180.52 C
I = total charge/time
Electrical charge carriers, often electrons or atoms deficient in electrons, travel as current. The current is frequently represented by the capital letter I. The standard unit is the ampere, represented by the letter A.
So I = 5034180.52/3600 = 1398.38 Amps
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a rope of mass m and length ` hangs from the ceiling. (i) what is the wave speed on the rope a distance y above the lower end?
A rope with the mass m and length L hangs from a ceiling. Show that the rope's wave speed at a distance of y above the lower end is v = gy.
What is the name for a wave's speed?
Wave speed is a feature of waves that can be used to describe the absolute value of phase velocity or the speed at which a wave phase moves at a specific frequency. group velocity, which differs from the phase velocity for dispersive waves, is the rate of propagation for the envelope of wave groups and frequently of wave energy.
What is a straightforward definition of wave velocity?
The distance a wave travels in a unit of time is measured by its wave velocity. It is the rate at which a particle is disturbed by something like a peak
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A 60kg board that is 6 m m long is placed at the edge of a platform, with 4m of its length extending over the edge. The board is held in place by blocks of masses M1M1 and M2M2 placed with their centers of mass on either end. If M2=30kg M2=30kg , what is the minimum value of M1 needed to keep the board from falling off the platform?
30kg
50kg
60kg
70kg
90kg
The minimum value of M1 to keep a board from falling off the platform is 90 kg.
The mass (M1) placed on the left side edge of the board
the mass of the board = 60 kg
the length of the board = 6 m
If the mass on the right side = 30 kg, and the length of the board L1 = 2m
Then, the length L2 which extend over the edge = 4m
Consider the center of gravity in the board that lies at the length of the board midpoint.
Then, the distance (D) of the gravity center from the platform end = 3 - 2
= 1
The moment when the platform end, the mass (M1) placed on the left side edge of the board can be computed as:
M1= 60kg+120kg/2
M1=180kg/2
M1=90kg
Therefore, The minimum value M1 needed to keep the board from falling off the platform is 90 kg.
Mass is a quantitative measure of inertia, a fundamental property of matter.
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a solid cylinder with a mass of 5 kg and a radius of 30 cm is rolling on a flat surface. the cylinder rolls with a speed of 2 m/s. what its kinetic energy?
The kinetic energy of the cylinder rolling is 10J
Rolling on a flat surface is a solid cylinder with a mass of 5 kg and a radius of 30 cm. 2 m/s is the rolling speed of the cylinder.
Due to frictional forces, when you slide an object over a surface (like a book over a table), it usually slows down quickly.
The frictional force acting provides the torque necessary for the object to begin rolling by causing it to change its angular momentum.
A moving object or particle has kinetic energy, which depends on both its mass and its rate of motion.
The type of motion could be vibration, rotation about an axis, translation (or motion along a path from one place to another), or any combination of these.
[tex]K.E=mv^2/2[/tex]
=[5 x 2 x 2]/2
=10 J
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An electric iron consumes energy at the rate of 420 W when heating is at the maximum rate and 180 W when heating is at the minimum rate. The applied voltage is 220 V. What is the current in each case?
when heating is at the minimum rate. The applied voltage is 220 V. The current in each case is 271.60 (approx)
P = VI V is voltage and I is current
then,
I = P/V
when heating is at maximum
current:
I = 420W/220V
=1.9(approx)
resistance:
R = V/I
= 220V/1.9A
=115.78(approx)
when heating in minimum
current:
I = P/V
= 180W/220V
= 0.81 (approx)
resistance:
R = V/I
=220V/0.81A
=271.60 (approx)
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the flux through a loop of wire changes at a uniform rate from 4.0x10-5 wb to 5x10-5 wb in 0.10 s. determine the emf induced in the coil
The emf induced in the coil is 4.0x10-4 V.
What is the emf induced in the coil?The induced emf in a coil is determined by multiplying the coil's number of turns by the opposite of the magnetic flux's rate of change. It involves the interaction of a magnetic field and a charge.The induced emf is equal to (BA cos) = − d/dt. The magnetic field's strength might fluctuate over time. The region that the loop encloses is subject to vary throughout time.A motional emf is an emf that is caused by movement in relation to a magnetic field. The length of the object moving at speed v in relation to the magnetic field's strength B is represented by the equation emf = LvB.Given data :
The induced emf is ε = - d/dt (BA cos θ).
d = 4.0x10-5Δd = 5x10-5t = 0.10 sApply the formula ,
induced emf is ε = - d/dt (BA cos θ).
= - 4.0x10-5 / 5x10-5 x 0.10 = 4.0x10-4 V.
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46. the sled is pulled up a higher hill that is less steep than the original hill described before question 44. how does the speed of the sled at the bottom of the hill (after it has slid down) compare to that of the sled at the bottom of the original hill?
Both hills' lowest speeds for the sled are the same.
Why would someone use a sled?Sleds—also known as sleighs—are wintertime vehicles that are often pulled across ice or snow by either horses or dogs. The travois and the sidecar, which were its predecessors, are said to have been the earliest human-powered vehicles.
What is the difference between a sled and a sleigh?An open-topped, often medium- to large-sized sleigh is used to transport people or cargo. Reindeer, dogs, or horses are frequently used as the drivers. While the name "sled" is still used generally in American parlance, it frequently refers to a smaller, frequently recreational, equipment.
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Nasa’s orion spacecraft, which left earth last week, flew within 81 miles of the surface of which celestial body on monday?.
Nasa’s Orion spacecraft, which left earth last week, flew within 81 miles of the surface of the moon on Monday .
What is the Orion spacecraft?The NASA Artemis program uses the crewed Orion spacecraft, which is largely reusable. The spacecraft is made up of an Airbus Defence and Space European Service Module and a Lockheed Martin Crew Module space capsule.This picture, obtained by a camera on the tip of one of Orion's solar arrays on the sixth day of the Artemis I mission, shows a significant chunk of the far side of the moon just beyond the Orion spacecraft.On Nov. 20, 2022, the spacecraft entered the lunar sphere of influence, making the moon the major gravitational force pulling on it rather than Earth. Its closest approach, which occurred on November 21, was at a distance of 80 miles from the lunar surface.Orion takes pictures of Earth and the moon at various phases and distances using its optical navigation camera.To learn more about Orion spacecraft, refer:
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what method would astronomers use to find the distance to a galaxy so far away that individual stars are impossible to make out (resolve)?
The method would astronomers use to find the distance to a galaxy so far away that individual stars are impossible to make out (resolve) is finding the redshift .
What is meant by finding redshift ?A manifestation of the Doppler Effect is redshift. The sound or light waves that are released by an object as it moves away from us are stretched out, which lowers their pitch and causes them to migrate towards the red end of the electromagnetic spectrum, where light has a longer wavelength.
An important topic for astronomers is "red shift." The term can be taken literally; when the light's wavelength is stretched, the light is perceived as having "shifted" toward the red portion of the spectrum. When a source of sound moves in relation to an observer, sound waves experience a similar phenomenon.
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. a 750 kg automobile is moving at 20.0 m/s at a height of 5.0 m above the bottom of a hill when it runs out of gasoline. the car coasts down the hill and then continues coasting up the other side until it comes to rest. ignoring frictional forces and air resistance, what is the value of h, the highest position the car reaches above the bottom of the hill?
The highest position the car reaches above the bottom of the hill is 20.9m.
Solution:
U = mgh
K= 1/2 mv2
Ki+Ui = Uf
750(0.5)(400)+750(9.8)(5)=(750)(9.8)h
200+4.9=9.8h
h = 20.9 m
Frictional force refers to the force created by two surfaces contacting and sliding against each other. Several factors affect frictional forces: These forces are primarily affected by surface textures and the forces that drive them together.
Scientists don't know exactly what causes friction. However, it is thought to be caused by interactions when small bumps on the surface rub against each other. The ridges on each surface flex and exert forces on each other making the surfaces less slippery against each other.
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You use j of energy to move a 2.0 N object. How far did you move it?
The distance the 2.0 N object will move, given that 35 J of energy is used to move it is 17.5 m
How do I determine the distance the object will move?Work done is defined as the product of force and the distance moved in the direction of the force. Mathematically, it is expressed as:
Work done (Wd) = force (F) × distance (d)
Using the above formula, we can detertmine the distance the object will move as illustrated below:
Workdone (Wd) = 35 JForce (F) = 2.0 NDistance moved (d) = ?Work done (Wd) = force (F) × distance (d)
35 = 2 × distance moved
Divide both sides by 2
Distance moved = 35 / 2
Distance moved = 17.5 m
Therefore, we can conclude from the calculation made above that the distance moved is 17.5 m
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Complete question:
You use 35 J of energy to move a 2.0 N object. How far did you move it?
a spring has a natural length of 12.0 cm. if a 26.0-n force is required to keep it stretched to a length of 26.0 cm, how much work w is required to stretch it from 12.0 cm to 19.0 cm? (round your answer to three decimal places.)
The work required to stretch the spring from 12 cm to 19 cm is 0.91 J
What is work done ?When an object is moved across a specific distance by an external force, work is the quantity of energy that is transmitted to the object.
According to the given information
Natural length = 12 cm
Stretched length = 26 cm
Change in length ( x ) = 26 - 12 = 14 cm = 0.14 m
Force required = 26 N
It is known to us that spring force F = K x
K = [tex]\frac{F}{x}[/tex]
K = [tex]\frac{26}{0.14}[/tex]
K = 185.7 N/m
This is the stiffness of the spring
Now the Force that is required to stretch the spring F = K[tex]x_{1}[/tex]
[tex]x_{1}[/tex] = change in length of the spring = 19 - 12 = 7 cm = 0.07 m
F = 185.7 × 0.07
F = 13 N
Now the work that is done in stretching the spring W = F × [tex]x_{1}[/tex]
W = 13 × 0.07
W = 0.91 J
The work required to stretch the spring from 12 cm to 19 cm is 0.91 J
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a ball is thrown upward with an initial velocity of 64 feet per second at an initial height of 80 feet. find the positio
The position of the object thrown upward with an initial velocity of 64 feet per second at an initial height of 80 feet from the ground after 10 seconds from the ground is 709.38 m
According to equation of motion,
s = u t + 1 / 2 a t²
s = Displacement
u = Initial velocity
t = Time
a = Acceleration
u = 64 ft / s
u = 19.5 m / s
t = 10 s
s = ( 19.5 * 10 ) + ( 1 / 2 * 9.8 * 10² )
s = 195 + 490
s = 685 m
Total distance from the ground,
d = h + s
h = 80 ft = 24.38 m
d = 24.38 + 685
d = 709.38 m
The final position of the object from the ground is 709.38 m
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a resistor and inductor are connected to a 9.0 v battery by a switch as shown. the moment the switch is closed, current flows through the circuit. the resistor has a resistance of r
The resistance of the resistor r so maximum current is 40.9 mA.
A Resistor and inductor are connected to a 9.0 v battery by a switch as shown. the moment the switch is closed, current flows through the circuit. the resistor has a resistance of r.
Calculation:-
Maximum current imax = V/R
= 9/220
= 40.9 mA.
Resistors are used for many purposes. a few examples consist of restricting electric powered cutting-edge, voltage division, warmth generation, matching and loading circuits, benefit manage, and setting time constants. they may be commercially available with resistance values over various extra than nine orders of magnitude.
A circuit is a entire circular direction that energy flows via. A easy circuit includes a modern supply, conductors and a load. The time period circuit may be utilized in a preferred sense to consult any fixed route that energy, records or a sign can travel via.
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at what speed do a bicycle and its rider, with a combined mass of 100 kg , have the same momentum as a 1800 kg car traveling at 4.8 m/s ?
Answer:
Explanation:
Given:
m₁ = 100 kg
m₂ = 1800 kg
V₂ = 4.8 m/s
____________
V₁ - ?
According to the condition of the problem, the momentum are equal:
m₁·V₁ = m₂·V₂
Speed:
V₁ = m₂·V₂ / m₁ = 1800·4.8 / 100 ≈ 86 m/s
force on moving charges: an electron moves with a speed of 8.0 × 106 m/???? along the x-axis. it enters a region where there is a magnetic field of 2.5 T, directed at an angle of 60° to the x ????x???????? and lying in the xy p????????????????. calculate the magnitude of:
The electron is subjected to a magnetic field accelerates at a rate of 3.0 E18 m/s.
What is a good example of the magnetic force?The magnets are attracted to or repelling one another as a result of this force. A compass, a motor, magnetic magnets that hold items in refrigerators, railroad tracks, and modern roller coasters are examples of objects that use magnetic force. A magnetic field is created by all moving charges, and any charges that move across its regions feel a force.
F=qvBsinθ
F=1.6×10−19 C×8.0×106m/s×2.5T×sin60∘
F=2.8×10−12 N
What is an example of a Class 8 magnetic force?A magnet exerts a magnetic pull on particular materials like metal, steel, nickel, and cobalt. This force can work from a distance, making it a sort of non-contact force. It does not needed that the objects be in direct contact. We learnt how to utilize a magnet to draw coins in the example above.
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While roller-skating, Granny collides with her tiny grandson Ambrose who is at rest. Ignoring …
a. Granny catches him and they both move together.
b. he and Granny make a bouncing collision, each going in opposite directions.
Momentum is conserved when two moving objects of equal masses collide while traveling in the same direction. The impact between cars 1 and 2 causes car 2 to accelerate.
According to the rule of conservation of momentum, absent an external force, the combined momentum of two or more bodies operating upon one another in an isolated system remains constant. As a result, momentum cannot be gained or lost. Collisions between two objects naturally fall under Newton's third law of motion. When two objects collide, forces of equal magnitude and opposite direction are applied to each object. These forces frequently result in one object gaining momentum and speeding up while the other object slows down.
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which term best describes the shape of jupiter's orbit around the sun?
Answer:
Explanation:
Kepler's First Law Describes the Shape of an Orbit
The orbit of a planet around the Sun (or of a satellite around a planet) is not a perfect circle. It is an ellipse—a “flattened” circle. The Sun (or the center of the planet) occupies one focus of the ellipse.
sunlight just outside earth's atmosphere has an intensity of 1.40 kw/m2. (a) calculate em for sunlight, assuming it to be a plane wave.
Electromagnetic waves for sunlight is 966.48 V/m
a) We know that the intensity of light
I = (1/2)ε0cEm2
From the above equation we can write
Em = √(2I / ε0c)
= √2 * 1.24 x 103 / 8.85 x 10-12 * 3 x108
= 966.48 V/m
When an electric field and a magnetic field vibrate together, electromagnetic waves, or EM waves, are produced. Thus, magnetic and electric fields oscillate to form electromagnetic waves (EM waves). When an electric field and a magnetic field interact, electromagnetic waves are produced. It follows that they are referred to as "electromagnetic" waves. An electromagnetic wave's electric and magnetic fields are parallel to one another and form a right angle. In addition, they are parallel to the EM wave's direction.
In vacuum, the speed of electromagnetic waves is constant at 3.00 x 108 ms-1. Neither the electric nor magnetic fields can deflect them. On the other hand, they can exhibit diffraction or interference.
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two balls having different speeds roll off the edge of a horizontal table at the same time. which hits the floor sooner, the faster ball or the slower one?
Two balls having different speeds roll off the edge of a horizontal table at the same time, both will hit the floor at the same time.
What is velocity?Velocity is defined in physics as a vector measurement of direction and velocity of motion. More specifically, the velocity of an object can also be defined as the rate of change of the object's position with respect to the reference frame and time. Simply put, the term velocity describes how fast an object moves in a particular direction. Indicates whether something is moving slowly or quickly. Let's look at an example where two objects are moving. If two objects are moving in the same direction, it's easy to tell which is moving faster. However, it is very difficult to determine which object is faster when both are moving in opposite directions. Considering such difficulties and ways to overcome them, our scientists developed the concept of velocity to make it easier for observers to identify faster or slower objects.
Both balls will hit the ground simultaneously. This is because both balls were moving parallel to the ground when they left the table (remember the table is horizontal). Both balls have 0 velocity in the direction of interest, both accelerate at the same velocity and land at the same time.
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what is the z -component of the electric field at a depth δ beneath the origin along the negative z -axis? express your answer in terms of some or all of the variables σ , r , δ , and the electric constant ϵ0 .
In this problem we have spherical symmetry, so we can apply Gauss theorem to find the magnitude of the electric field:
∫ E(r) . dr = q / ε₀
where the term on the left is the flux of the electric field through the gaussian surface, and q is the charge contained in the surface.
Here we are analyzing the field at a distance , so outside the solid ball. If we take a gaussian sphere with radius r, we can rewrite the equation above as:
E(r) . 4πr² = q/ε₀ (1)
where 4πr² is the surface of the sphere.
The charge contained in the sphere, q, is equal to the charge density times the volume of the solid ball, 4/3πr³ₐ:
q = ρ (4/3πr³ₐ) (2)
Combining (1) and (2), we find
E(r) . 4πr² = (4/3ρπr³ₐ) / (3ε₀)
E(r) = ρπr³ₐ / 3ε₀r²
And we see that the electric field strength is inversely proportional to the square of the distance, r.
B) Now we are inside the solid ball: . By taking a gaussian sphere with radius r, the Gauss theorem becomes
E(r) . 4πr² = q / ε₀ (1)
But this time, the charge q is only the charge inside the gaussian sphere of radius r, so
q = ρ(4/3πr³) (2)
Combining (1) and (2), we find
E(r) . 4πr² = 4ρπr³ / 3ε₀
E(r) = ρr / 3ε₀
And we see that this time the electric field strength is proportional to r.
C) E(0)=0.
limr→∞E(r)=0.
The maximum electric field occurs when r=rb.
From part A) and B), we observed that
- The electric field inside the solid ball (r <r ₐ) isρr / 3ε₀ (1)
so it increases linearly with r
- The electric field outside the solid ball ( r > rₐ) isE(r) = ρrₐ³ / 3ε₀r² (2)
so it decreases quadratically with r
--> This implies that:
1) At r=0, the electric field is 0, because if we substitute r=0 inside eq.(1), we find E(0)=0
2) For r→∞, the electric field tends to zero as well, because according to eq.(2), the electric field strength decreases with the distance r
3) The maximum electric field occur for , i.e. on the surface of the solid ball: in fact, for the electric field increases with distance, while for the field decreases with distance, so the maximum value of the field is for .
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[NOTE: THIS IS AN INCOMPLETE QUESTION. THE COMPLETE QUESTION IS: A solid ball of radius rb has a uniform charge density ρ.
Part A What is the magnitude of the electric field E(r) at a distance r>rb from the center of the ball? Express your answer in terms of ρ, rb, r, and ϵ0.
Part B What is the magnitude of the electric field E(r) at a distance r Express your answer in terms of ρ, r, rb, and ϵ0.
Part CLet E(r) represent the electric field due to the charged ball throughout all of space. Which of the following statements about the electric field are true?
Check all that apply.
E(0)=0.
E(rb)=0.
limr→∞E(r)=0.
The maximum electric field occurs when r=0.
The maximum electric field occurs when r=rb.
The maximum electric field occurs as r→∞.]
A 100-g firecracker is launched vertically into the air and explodes into two pieces at the peak of its trajectory. If a 72-g piece is projected horizontally to the left at 20 m/s, what is the speed and direction of the other piece?.
The second particle would therefore move at a speed of (-51.5) m/s.
What is Velocity ?Velocity is the directionally speed of an object in motion, as measured by a certain unit of time and seen from a particular point of reference.
According to the given information
Mass (m) = 100 gm
20 m/s is the speed.
Vertical velocity is zero at its maximum.
The system's horizontal momentum is conserved because there is no horizontal force acting on it.
[tex]m_{2}[/tex] = 100-72= 28g
[tex]m_{1}v_{1} + m_{2}v_{2}[/tex] = 0
[tex]v_{2} = -\frac{m_{1}v_{1} }{m_{2} }[/tex]
[tex]v_{2}[/tex] = (72×20)/(28)
[tex]v_{2}[/tex] = -51.5 m/s
The second particle would therefore move at a speed of (-51.5) m/s.
Negative signs point in the direction of rightward motion.
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