physics exam 3
It depends on the axis of rotation and how mass is distributed within an object
For an isolated system, which statement(s) are true about the rotational kinetic energy of a rigid object rotating around an axis of symmetry (check all that are true)? 1. it depends on the direction of the angular velocity of the object 2. it is a conserved quantity 3. it depends on the axis of rotation 4. it depends on how mass is distributed within an object *remember: energy is a scalar*
the initial angular speed is smaller
Initially, a figure skater spins frictionlessly on ice with their arms outstretched, away from their body. The skater then pulls their arms in and holds them against their body. How does the initial and final angular speeds of the skater compare? *The system of the person and the wheel is isolated and experiences no external torques. Angular momentum must therefore be conserved. The skater has a larger moment of inertia when their arms are outstretched.*
1kg
A 1-kg rock is suspended by a massless string from one end of a uniform 1-m measuring stick. What is the mass of the measuring stick if it is balanced by a support force at the 0.25-m mark?
20 cm
A block attached to a spring is oscillating with an amplitude of 10cm. What will be the block's amplitude if its total energy is quadrupled? *The maximum amplitude occurs when all of the oscillator's kinetic energy has been converted into potential energy.*
The block because it has a moment of inertia of zero
A block of ice and a disk are released from rest at the same vertical elevation. The block slides with almost no friction down a hill and the disk rolls down the hill. Which object will reach the bottom first and why?
28 cm/s
A block oscillating on a spring has a maximum speed of 20 cm/s. What will the block's maximum speed be if the total energy is doubled? *Energy is conserved in a simple harmonic oscillator as kinetic energy is transferred into potential energy and then back to kinetic energy. The maximum speed occurs when the potential energy is zero.*
1/2mg
A board of mass m lies on the ground. What is the magnitude of the force you would have to exert to lift one end of the board barely off the ground (assuming that the other end still touches the ground)? *If the board is uniform, its center of mass will be in its middle and the gravitational force acts through the center of mass. You must exert a torque that is equal and opposite to this to lift the board.*
to your right
A cylinder rolls without slipping directly toward you down an incline. From your perspective, in which direction does the cylinder's angular acceleration point?
to your right
A cylinder rolls without slipping directly toward you down an incline. From your perspective, in which direction does the cylinder's angular velocity point? *Use the right hand rule to curl your fingers in the direction the cylinder is rolling. Your thumb will point in the direction of the angular velocity.*
same for both
A force F is applied to a dumbbell for a time interval ∆t, first as in (a) and then as in (b). In which case does the dumbbell acquire the center of mass speed?
b
A force F is applied to a dumbbell for a time interval ∆t, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater energy?
zero
A friend is running clockwise at a constant speed on a circular track. At the instant your friend is located due west of you, their velocity is in the direction pointing due east. What is the direction of your friend's angular momentum around you at that instant? *The angular momentum of an object moving past a point is the cross product of the distance from the axis to the object crossed into the linear momentum. If the vectors are parallel, there is no angular momentum.*
6 cm
A glider on an air track is connected by a spring to the end of the air track. If it takes 0.30 s for the glider to travel the distance of 12 cm from one turning point to the other, what is the glider's amplitude? *the amplitude of an oscillator is the distance from the equilibrium position to it's maximum or minimum value*
10.5 s^-1
A glider on an air track is connected by a spring to the end of the air track. If it takes 0.30 s for the glider to travel the distance of 12 cm from one turning point to the other, what is the glider's angular frequency?
North
A jet is cruising at a constant velocity due east. At the instant the jet is above your head, what is the direction of the jet's angular momentum around you? *The angular momentum of an object moving past a point is the cross product of the distance from the axis to the object crossed into the linear momentum.*
the tension is greater than the object's weight
A weight hangs from a massless string but is pulled to one side by a horizontal massless string, as shown. How does the tension force exerted by the angled string compare to the hanging object's weight? *The force exerted by the angled string must be cancel the sum of the gravitational force acting on the block and that exerted by the horizontal string.*
figure 1
An amusement park ride spins around and then has the floor drop out from under the riders' feet. A rider on this ride then finds themselves stuck with their back to the wall and their feet not touching the floor. Which diagram correctly shows the forces acting on the rider?
Move speaker 2 forward (right) 0.5m
Two loudspeakers emit sound waves with the same wavelength and the same amplitude. Which of the following would cause there to be constructive interference at the position of the dot? *For constructive interference to happen, we need the waves at the point to be exactly in phase. Note that wave 1 at the dot has zero displacement and positive slope. Where is the nearest point on wave 2 with zero displacement and positive slope? What does that imply about how we should move the speaker?*
Move speaker 2 backward (left) 0.5m
Two loudspeakers emit sound waves with the same wavelength and the same amplitude. Which of the following would cause there to be destructive interference at the position of the dot? *Correct! Note that wave 1 has zero displacement and positive slope at the dot. For destructive interference to happen, you need the wave from speaker 2 to also have zero displacement but negative slope. You can do this by moving speaker 2 to the left by 1/4 of the wavelenght (0.5m).*
0.5m
Two speakers placed along the x axis produce waves with a 1.0m wavelength that are exactly out of phase when they are created. You want a person to the right of both speakers to experience constructive experience. Which of the options below would make this possible? *Correct! Separating the speakers by half a wavelength would cause an additional phase difference of 180°, making the two speakers be in phase again.*
softer sound
Two speakers produce waves of the same wavelength that are exactly out of phase when they are created. At a point midway between the speakers, you would expect to hear *Correct! The total phase difference is the sum of the initial phase difference (180°) with the phase difference coming from the path length (0°), so the waves interfere destructively.*
louder sound
Two speakers produce waves of the same wavelength that are in phase. At a point midway between the speakers, you would expect to hear *Correct! Since the two speakers are in phase, their sound waves will interfere constructively at the midpoint since there's no path length difference.*
wavelength, propagation speed
Two strings, one thick and one thin, are connected to form one long string. A wave travels along the string and passes the point where the two strings are connected. Which of the following change(s) at that point? *The propagation speed depends on the physical properties of the medium. If these change, so will the propagation speed. In addition, any two adjacent parts of the string must always move at the same frequency, regardless of their other properties.*
the wheel with spokes
Two wheels have the same total mass M and radius R. One wheel is a uniform disk. The other is like a bicycle wheel, with lightweight spokes connecting the rim to a small hub. Which has the larger moment of inertia?
arrangement 3
You are using a wrench to try and loosen a bolt. Which of the arrangements shown below is the least effective in loosening the bolt? *The magnitude of the total torque is the force times the distance from the axis of rotation to the point where the force is applied times the sine of the angle between them. For a given value of force and distance, the largest torque will occur when the force is applied perpendicular to the radius (with an angle of 90∘).*
arrangement 2
You are using a wrench to try and loosen a bolt. Which of the arrangements shown below is the most effective in loosening the bolt?
it stays about the same
You stand 40 m away from a friend, facing them. The friend throws a 2 N ball vertically upwards over their head at a speed of 8 m/s. As the ball continues to move upwards, how does the magnitude of the gravitational torque on the ball change? *Remember that in the torque equation, only the distance perpendicular to the force matters; this what the mathematical operation of the cross product does.*
to your left
You stand 40 m away from a friend, facing them. The friend throws a 2 N ball vertically upwards over their head at a speed of 8 m/s. What is the direction of the gravitational torque on the ball, relative to you, just after it is thrown? *Use the right hand rule to find the direction in a cross product. The radius ALWAYS points in the direction that goes from the axis of rotation out to the place where the force acts.*
80 Nm
You stand 40 m away from a friend, facing them. The friend throws a 2 N ball vertically upwards over their head at a speed of 8 m/s. What is the magnitude of the gravitational torque on the ball, relative to you, just after it is thrown?
Energy
_________ is conserved in an (ideal) standing wave. The potential energy of a wave depends on its amplitude. The kinetic energy depends on the change of the particle's position.
Torque
a force that causes rotation a measure of the force that can cause an object to rotate about an axis. Just as force is what causes an object to accelerate in linear kinematics, torque is what causes an object to acquire angular acceleration. Torque is a vector quantity.
standing waves
antinodes and nodes occur in.... *nodes and antinodes result in the interference of two waves moving in opposite directions*
28 cm
A wave is traveling on a string at 200cm/s at a frequency of 10Hz. If the frequency is kept constant, what is the wavelength if the string tension is doubled?
doppler effect
-the person moving away from the source means they'll hear a lower frequency, so the numerator needs to get smaller (hence the minus sign) -the source moving away from the person also means the person will hear a lower frequency, so the denominator needs to get larger (hence the plus sign)
figure C
A non-uniform box has a center-of-mass that is off-center, as indicated by the dot in the figure below. The boxes are each placed on an inclined plane. In which of the four orientations shown does the box tip over?
figure b and a
A particle rotates around a circle on a massless, rigid rod. Which figure(s) shown below has the largest angular acceleration?
figure d and c
A particle rotates around a circle on a massless, rigid rod. Which figure(s) shown below has the smallest angular acceleration? *Angular acceleration is the torque divided by the moment of inertia. The force, radius, and mass of the particle all determine the angular acceleration.
the person rotates counterclockwise (as seen from above)
A person is sitting at rest on a stool that is free to frictionlessly rotate about a vertical axis. The person holds bicycle wheel in one hand that is rapidly spinning counterclockwise (when viewed from above). The person then stops the wheel with the other hand. What happens as a result? *The system of the person and the wheel is isolated and experiences no external torques. Angular momentum must therefore be conserved.*
initial frequency
A sound wave with wavelength (lambda) and frequency f0 moves into a new medium in which the speed of sound is v1=2v0. What is the new frequency?
1/2 initial wavelength
A sound wave with wavelength (lambda) and frequency f0 moves into a new medium in which the speed of sound is v1=2v0. What is the new wavelength?
Direction B
A student holding a heavy ball sits on the outer edge a merry go round which is initially rotating counterclockwise, as seen in the figure below. Which way should the student throw the ball so that they increase their rotation rate? *Angular momentum is conserved. If the ball has angular momentum in the opposite direction to angular momentum of the rotating person, the person must rotate faster to compensate. The ball has angular momentum as it moves past the rotation point in the center of the merry go round.*
Direction A
A student holding a heavy ball sits on the outer edge a merry go round which is initially rotating counterclockwise, as seen in the figure below. Which way should the student throw the ball so that they stop the rotation? *Angular momentum is conserved. If the ball has angular momentum in the same direction as the angular momentum the rotating person initially has, the person will stop rotating to compensate. The ball has angular momentum as it moves past the rotation point in the center of the merry go round.*
rotational version of newton's 2nd law
According to Newton's Second Law, Στ = I α
angular momentum
An isolated star collapses so that its radius is half its original radius. What quantities are conserved in this collapse? *The system is isolated, so there are no external torques and angular momentum is therefore conserved. Mechanical energy is also conserved.*
Doppler effect
An observed change in the frequency of a wave when the source or observer is moving
Disk B
Disk A has a wooden center surrounded by iron. Disk B has an iron center surrounded by wood. The two disks are equal in their shape, size, and mass. The two disks are released from rest at the same time and vertical position and roll without slipping down an incline. Which disk arrives at the bottom of the hill first? *the moment of inertia increases when there is more mass far from the axis of rotation*
Disk 2 has twice the angular momentum of disk 1
Disks 1 and 2 (shown below) have equal mass. Disk 2 has double the radius and half the angular speed of Disk 1. How does the angular momentum of disks 1 and 2 compare? *For an object rotating around its own center of mass, the angular momentum depends on the moment of inertia and the angular velocity.*
Slightly increase the length of a day
If global warming continues as anticipated, the polar ice caps may melt, substantially raising sea levels around the world. What affect would this have on the duration of the day? *Water will move away from the poles and increase the Earth's moment of inertia. Earth is isolated, so its angular momentum will be conserved, thus affecting the rotation rate of the Earth.*
increases by a factor of sqrt(2)
If you double the spring constant of a harmonic oscillator, the oscillation frequency does what? *Frequency and angular frequency are directly proportional to one another.*
it is all in the form of kinetic energy
If you take snapshots of a standing wave on a string, there are certain instants when the string is totally flat. What has happened to the energy of the wave at those instants?
same for both
In Case 1, a force F is pushing perpendicular on an object a distance L/2 from the rotation axis. In Case 2 the same force is pushing at an angle of 30 degrees a distance L from the axis. In which case is the torque due to the force about the rotation axis biggest?
they are equal
In Case 1, one end of a horizontal plank of mass M and length L is attached to a wall by a hinge and the other end is held up by a wire attached to the wall. In Case 2 the plank is half the length but has the same mass as in Case 1, and the wire makes the same angle with the plank. In which case is the tension in the wire biggest? *The vector sum of the hinge, the plank's weight, and the tension must be zero. The board has the same mass in both cases.*
case 1
In Case 1, one end of a horizontal plank of mass M and length L is attached to a wall by a hinge and the other end is held up by a wire attached to the wall. In Case 2 the plank is half the length but has the same mass as in Case 1, and the wire makes the same angle with the plank. In which case is the torque due to the tension in the wire the biggest?
sin(kx) and cos(kx) satisfy their respective boundary conditions at x=0 with no phase needed
In calculating the standing wave patterns in tubes of air of length L, why is it convenient to use cos(kx) for the case of the tube that is open on both sides and sin(kx) in the case that the tube is closed at the x=0 side and open at the x=L side?
decrease the length by a factor of 2
In music, two notes are said to be an octave apart when one note is exactly twice the frequency of the other. Suppose you have a guitar string playing frequency f0. To increase the frequency by an octave, to 2f0, by what factor would you have to change the length? [Note: guitar players achieve this by moving their fingers to various frets to create a node there.] *An integer number of half wavelengths must still fit between the length of the string. The wave speed and the mode won't change in this process.*
4 J/m^2
See below for a kinetic energy graph of a 16kg mass oscillating on a horizontal spring. What is the spring constant for this spring?
0.5 rad/sec
See below for a kinetic energy graph of a 16kg mass oscillating on a horizontal spring. With what frequency does this mass oscillate?
at t=1, 3, 5, and 7 sec
See the trace below for a particle undergoing periodic motion. At what times is the particle instantaneously at rest?
disk 1
Strings are wrapped around the circumference of two solid disks and pulled with identical forces. Disk 1 has a bigger radius, but both have the same moment of inertia. Which disk has the biggest angular acceleration?
string C
Suppose an object is suspended from strings as shown in the diagram below. Which exerts the largest force on the connecting ring? *The forces exerted by strings A and B must cancel the force exerted by C if the ring is to remain at rest. The vertical components of A and B add to cancel with the force exerted by C and the horizontal components must cancel with each other.*
string A
Suppose an object is suspended from strings as shown in the diagram below. Which exerts the smallest force on the connecting ring? *The forces exerted by strings A and B must cancel the force exerted by C if the ring is to remain at rest. The vertical components of A and B add to cancel with the force exerted by C and the horizontal components must cancel with each other.*
ci < cf
Suppose we take a disk with initial radius R, initial mass M, and initial moment of inertia Ii=ciMR^2, where ci is the constant of proportionality. We then uniformly remove material from the center of the disk, creating a hole at the object's center. The new object has an inner radius r<R , outer radius R, and mass m<M. How does the constant of proportionality cf in the moment of inertia of the new object If compare to the constant of proportionality for the original disk? *because once the material is removed, the object is no longer a disk*
ci = cf
Suppose we take a disk with initial radius R, initial mass M, and initial moment of inertia Ii=ciMR^2, where ci is the constant of proportionality. We then uniformly remove material from the outer part of the disk. The new disk has an inner radius r<R , outer radius R, and mass m<M. How does the constant of proportionality cf in the moment of inertia of the new object If compare to the constant of proportionality for the original disk? *the object is still a disk*
it decreases
Suppose you are looking down on a disk that is spinning clockwise. If you apply an upward torque to the disk, what happens to the angular speed? *Torque is the rotational equivalent of force in the rotational version of Newton's 2nd Law. Torque causes changes to angular acceleration. A torque in the same direction as the object is already spinning will cause it to speed up in the same way a force in the same direction an object is already moving linearly will cause it to speed up.*
it increases
Suppose you are looking down on a disk that is spinning counterclockwise. If you apply an upward torque to the disk, what happens to the angular speed? *Torque is the rotational equivalent of force in the rotational version of Newton's 2nd Law. Torque causes changes to angular acceleration. A torque in the same direction as the object is already spinning will cause it to speed up in the same way a force in the same direction an object is already moving linearly will cause it to speed up.
there are no standing waves at this frequency
Suppose you have a standing wave on a string between two boundaries separated by a length L: Suppose you double the tension but hold the frequency and separation distance between the boundaries fixed. What are the wavelengths of the standing waves now?
increases by a factor of sqrt(2)
Suppose you have a string with tension T and mass density (mew). How does the wavenumber of a traveling wave on this string change if the mass density is doubled but the frequency held constant?
8
The figure below shows a standing wave oscillating on a string at frequency f0. How many antinodes will there be if the frequency is doubled to 2f0? *The frequency and wavelength are related by the wave speed. How does the number of wavelength relate to the distance between the fixed ends?*
4
The figure below shows a standing wave oscillating on a string at frequency f0. What mode (value of m) is this? *The mode is determined by counting the number of antinodes.*
6.5J
The figure below shows the potential-energy diagram and the total energy line (TE) of a particle oscillating on a spring. What is the particle's maximum kinetic energy?
arrow D
The lid of a grand piano is propped open, as shown below. Which arrow most closely approximates the direction of the force that the hinge exerts on the lid? Assume that the prop exerts a force parallel to the prop's length (i.e., upwards and to the left).
It stays the same
The moment of inertia of a hollow spherical shell of mass M and radius R about its axis of symmetry is 2/3MR^2. A small amount mass (m<<M and size <<R) is added at one spot along the axis of rotation of the shell. How does the moment of inertia if the shell change when the small mass is added?
the way the mass is distributed within the object, the total mass of the object, the axis around which the object rotates
The moment of inertia of an object depends on...
3
The picture shows a displacement standing sound wave in a 32-mm-long tube of air that is open at both ends How many pressure nodes (x positions in tube where the pressure in the tube equals atmospheric pressure) are there?
2
The picture shows a displacement standing sound wave in a 32-mm-long tube of air that is open at both ends. If we call the fundamental mode m = 1, which mode (value of m) standing wave is this?
propagation speed
The propagation speed depends on the physical properties of the medium. If these change, so will the propagation speed. In addition, any two adjacent parts of the string must always move at the same frequency, regardless of their other properties.
they are all equal
The three wave pulses shown below travel along the same string. Which pulse has the largest wave speed?
pair 1
The traces below show beats that occur when two different pairs of waves are added.For which of the two is the difference in frequency of the original waves greater? *Remember that the beat frequency is not the frequency of oscillation of a particle, but rather the frequency at which the shape of the wave as a whole changes (you can think of this as the frequency that the amplitude is changing). The beat frequency is equal to the difference in frequency between the two waves.*
Pair 2
The traces below show beats that occur when two different pairs of waves are added.Which of the two pairs of original waves contains the wave with the highest frequency? *Correct! The frequency of oscillation within the beats themselves is the average of the two waves being added. Pair 2 has faster oscillations within the beats so it must be composed of higher frequencies.*
mass density, tension
The wave speed depends only on the physical properties of the medium. For a string, this is the ______ _______ of the string and the string's ________.
the frequency of sound is highest at C
Three stationary observers, A, B, and C are listening to a moving source of sound. The diagram below shows the location of the wave crests of the moving source with respect to the three observers. Which of the following is true? *Correct! Since all the wave crests move at the same speed, the wave frequency will be determined by the spacing between the crests that you observe at a point (f=v/λ), and that is highest at point C.*
force, perpendicular
To increase torque, you can either increase the _____ or increase the ________ distance from the axis of rotation to the point where the force is applied.
forces
Torque depends on the distance at which the _______ are applied.
1, 3
Two identical, short pulses of opposite amplitude travel along a stretched string and interfere destructively. Select all of the true statements about this situation. 1. there is a point on the string that does not move up or down 2. there are several points on the string that do not move up or down 3. there is an instant at which the string is completely straight 4. when the two pulses interfere, the energy of the pulses is momentarily zero
2N
Two wheels rotate around a fixed central axis. Each wheel has a mass of 1 kg and starts from rest. Forces are then applied as shown below. Assume the hubs and spokes are massless, so that the rotational inertia is that of a hoop (I=MR^2). In order to impart identical angular accelerations, how large must F2 be? *According to the rotational version of Newton's 2nd Law, angular acceleration is caused by the torque exerted on the wheel . For each wheel, the torque is equal to the angular acceleration times the moment of inertia.*
the material properties of the string
Upon what does the speed of propagation for a traveling wave on a string depend?
object 3
Which object shown below has the largest moment of inertia about the midpoint of the rod?
A
Which third force on the wheel, applied at point P, will make the net torque zero? For scale, vector C has the same length as the horizontal force.
Zero
You are standing 30 m due east of a 50 kg person who is running at a speed of 20 m/s due west. What is the magnitude of that person's angular momentum about you (in units of (kgm^2)/s)? *The angular momentum of an object moving past a point is the cross product of the distance from the axis to the object crossed into the linear momentum. If the vectors are parallel, there is no angular momentum.*
upward
You are standing due east of a car travelling southwest. What is the direction of the car's angular momentum around you? *The angular momentum of an object moving past a point is the cross product of the distance from the axis to the object crossed into the linear momentum.*
it stays the same
You are trying to open a door that is stuck by pulling on the doorknob in a direction perpendicular to the door. If you instead tie a rope to the doorknob and then continue to pull in the same direction (i.e., perpendicular to the door) with the same force, how does the torque you exert on the door change? *The torque is determined by the magnitude of the force, the distance from the axis of rotation to the place the force is applied, and the sine of the angle between them. The sine of this angle mathematically picks out only the distance perpendicular to the force.*
directly proportional
frequency and angular frequency are _________ _________ to one another
distribution of mass
the moment of inertia depends on the ______ around the axis of rotation
larger
the object with more mass distributed far from the center will have a ______ moment of inertia
moment of inertia
the resistance to rotation *an object's "resistance" to angular acceleration. It is the rotational analogue of mass.* I=L/w Inertia = angular momentum/angular velocity
torque
the rotational equivalent of force in the rotational version of Newton's 2nd Law
