7/24/2019 Mastering Physics HW 5 Ch 14 - Oscillations http://slidepdf.com/reader/full/mastering-physics-hw-5-ch-14-oscillations 1/13 HW 5 Ch 14 - Oscillations Due: 11:59pm on Tuesday, October 13, 2015 To understand how points are awarded, read the Grading Policy for this assignment. PhET Tutorial: Masses & Springs Learning Goal: To understand how the motion and energetics of a weight attached to a vertical spring depend on the mass, the spring constant, and initial conditions. For this tutorial, use the PhET simulation Masses & Springs. You can put a weight on the end of a hanging spring, stretch the spring, and watch the resulting motion. Start the simulation. When you click the simulation link, you may be asked whether to run, open, or save the file. Choose to run or open it. You can drag a weight to the bottom of a spring and release it. You can put only one weight on any spring. With the weight on the spring, you can click an drag the weight up or down and release it. Adjusting the friction slider bar at top right increases or decreases the amount of thermal dissipation (due to a resistance and heating of the spring). You can adjust the spring constant of spring #3 using the softness spring 3 slider bar. The horizontal dashed line well as the ruler can be dragged to any position, which is helpful for comparing positions of the springs. Feel free to play around with the simulation. When you are done and before starting Part A, set the friction slider bar to the middle and the gravitational acceleration back to "Earth". Part A Place a 50 weight on spring #1, and release it. Eventually, the weight will come to rest at an equilibrium position, with the spring somewhat stretch compared to its original (unweighted) length. At this point, the upward force of the spring balances the force of gravity on the weight. With the weight in its equilibrium position, how does the amount the spring is stretched depend on the mass of the weight? ANSWER: Correct Since the force of gravity on the weight increases as the mass increases, the upward force of the spring must increase for the two forces to balance (and the weight to therefore be in equilibrium). The force the spring exerts on the weight increases the more the spring is stretched from its unweighted length. Part B Use the simulation to estimate the masses of the three colored, unlabeled weights. Then, place them into the appropriate mass bins. Hint 1. How to approach the problem You learned in Part A that a heavier weight stretches the spring more at the equilibrium position. Compare the equilibrium positions for each of the colored weights to those for the weights with labeled masses. You might want to verify that the three springs all stretch the same amount for a particular weight, so you can compare the equilibrium positions simultaneously. The spring stretches less for a heavier weight. The spring stretches more for a heavier weight. The stretch does not depend on mass. Typesetting math: 100%
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To understand how points are awarded read the Grading Policy for this assignment
PhET Tutorial Masses amp Springs
Learning Goal
To understand how the motion and energetics of a weight attached to a vertical spring depend on the mass the spring constant and initial conditions
For this tutorial use the PhET simulation Masses amp Springs You can put a weight on the end of a hanging spring stretch the spring and watch the
resulting motion
Start the simulation When you click the simulation link you may be asked whether to run open or save the file Choose to run or open it
You can drag a weight to the bottom of a spring and release it You can put only one weight on any spring With the weight on the spring you can click an
drag the weight up or down and release it Adjusting the friction slider bar at top right increases or decreases the amount of thermal dissipation (due to a
resistance and heating of the spring) You can adjust the spring constant of spring 3 using the softness spring 3 slider bar The horizontal dashed line
well as the ruler can be dragged to any position which is helpful for comparing positions of the springs
Feel free to play around with the simulation When you are done and before starting Part A set the friction slider bar to the middle and the gravitational
acceleration back to Earth
Part A
Place a 50 weight on spring 1 and release it Ev entually the weight will come to rest at an equilibrium position with the spring somewhat stretchcompared to its original (unweighted) length At this point the upward force of the spring balances the force of gravity on the weight
With the weight in its equilibrium position how does the amount the spring is stretched depend on the mass of the weight
ANSWER
Correct
Since the force of gravity on the weight increases as the mass increases the upward force of the spring must increase for the two forces to
balance (and the weight to therefore be in equilibrium) The force the spring exerts on the weight increases the more the spring is stretched fromits unweighted length
Part B
Use the simulation to estimate the masses of the three colored unlabeled weights Then place them into the appropriate mass bins
Hint 1 How to approach the problem
You learned in Part A that a heavier weight stretches the spring more at the equilibrium position Compare the equilibrium positions for each of
the colored weights to those for the weights with labeled masses You might want to verify that the three springs all stretch the same amount
for a particular weight so you can compare the equilibrium positions simultaneously
A greater mass results in a lower frequency and a longer period of oscillation
Part H
The amplitude of oscillation is the maximum distance between the oscillating weight and the equilibrium position Determine the frequency of
oscillation for several different amplitudes by pulling the weight down different amounts
How does the frequency depend on the amplitude of oscillation
ANSWER
CorrectEven though the weight has to travel farther each oscillation if the amplitude is greater the spring on average exerts a stronger force causing a
greater acceleration and a greater average speed The effects of the longer distance and faster speed cancel out so that the period of oscillation
doesnrsquot change
Part I
The spring constant of spring 3 can be adjusted with the softness spring 3 slider bar (harder means a greater spring constant or stiffer spring)
How does the frequency of oscillation depend on the spring constant
Hint 1 How to approach the problem
You can place a weight on spring 3 with the softness of spring 3 set to a very low (soft) value and place another weight with the same masson spring 1 Then release the two weights and compare their frequencies of oscillation
ANSWER
Correct
It turns out that the frequency of oscillation depends on the square root of the ratio of the spring constant to mass where
is the frequency A stiffer spring constant causes the frequency to increase Sports cars use stiff springs whereas large plush Cadillacs use softsprings for their suspension
PhET Interactive Simulations
University of Colorado
httpphetcoloradoedu
PhET Tutorial Pendulum Lab
Learning Goal
To understand the relationships of the energetics forces acceleration and velocity of an oscillating pendulum and to determine how the motion of a
pendulum depends on the mass the length of the string and the acceleration due to gravity
The frequency increases as the mass increases
The frequency is independent of the mass
The frequency decreases as the mass increases
The frequency increases as the amplitude increases
The frequency decreases as the amplitude increases
The frequency is independent of the amplitude
The frequency increases as the spring constant increases
The frequency is independent of the spring constant
The frequency decreases as the spring constant increases
For this tutorial use the PhET simulation Pendulum Lab This simulation mimics a real pendulum and allows you to adjust the initial position the mass
and the length of the pendulum
Start the simulation You can drag the pendulum to an arbitrary initial angle and release it from rest You can adjust the length and the mass of the
pendulum using the slider bars at the top of the green panel Velocity and acceleration vectors can be selected to be shown as well as the forms of
energy
Feel free to play around with the simulation When you are done click the Reset button
Part A
Select to show the energy of pendulum 1 Be sure that friction is set to none
Drag the pendulum to an angle (with respect to the vertical) of and then release itWhen the pendulum is at what form(s) of energy does it have
Check all that apply
ANSWER
Correct
The pendulum starts off with no kinetic energy since it is released from rest so it initially only has potential energy When the pendulum is at
it is just as high above the ground as when it started so it must have the same amount of potential energy as it initially had Since thetotal energy is conserved it canrsquot have other forms of energy at (if it did it would have more energy there than it initially had) so it again has
only potential energy
Part B
Drag the pendulum to an angle (with respect to the vertical) of and then release it
Where is the pendulum swinging the fastest
ANSWER
Correct
The pendulum has the least potential energy at this location since it is at the lowest point in the arc (in fact for this simulation the potential
energy reference location is here so it has no potential energy) This means that the kinetic energy is greatest here so the pendulum is moving
Drag the pendulum to an angle (with respect to the vertical) of and then release it Select to show the acceleration vector
With the pendulum swinging back and forth at which locations is the acceleration equal to zero
ANSWER
Correct
The pendulum is moving in a circular path so its velocity is never constant In fact for most locations the acceleration has both a radial
component (the centripetal acceleration which is directed along the rope) and a tangential component (due to the speed changing directed along
the path the only place the tangential acceleration is zero is when the angle is )
Part D
With the pendulum swinging back and forth how does the tension of the rope compare to the force of gravity when the angle is
Hint 1 How to approach the problem
Look at the direction of the acceleration when the angle is (you can slow down or pause the simulation to see this more clearly) and think
about the relationship between the net force acting on the pendulum and the acceleration (Newtonrsquos 2nd law of motion) It would probably help todraw a free-body diagram for the mass
ANSWER
Correct
Since the acceleration of the pendulum is directed up when the angle is the net force must be directed up (Newtonrsquos 2nd law) This meansthat the upward force of tension must be stronger than the downward force of gravity
Part E
Drag the pendulum to an angle (with respect to the vertical) of and then release it
With the pendulum swinging back and forth where is the tension equal to zero
ANSWER
Correct
At these locations the acceleration is solely due to gravity and directed downward Thus the net force acting on t he pendulum i s also directed
downward meaning there are no horizontal forces This requires the tension to be zero
Part F
Now for parts F-I you will investigate how the period of oscillation depends on the properties of the pendulum
The period of oscillation is the amount of time it takes for the pendulum to take a full swing going from the original angle to the other side and
3 0
∘
The acceleration is zero when the angle is either or
The acceleration is zero when the angle is
The acceleration is never equal to zero as it swings back and forth
+ 3 0
∘
minus 3 0
∘
0
∘
0
∘
0
∘
0
∘
The tension is equal to the force of gravity
The tension is greater than the force of gravity only if it is swinging really fast
A greater mass results in a lower frequency and a longer period of oscillation
Part H
The amplitude of oscillation is the maximum distance between the oscillating weight and the equilibrium position Determine the frequency of
oscillation for several different amplitudes by pulling the weight down different amounts
How does the frequency depend on the amplitude of oscillation
ANSWER
CorrectEven though the weight has to travel farther each oscillation if the amplitude is greater the spring on average exerts a stronger force causing a
greater acceleration and a greater average speed The effects of the longer distance and faster speed cancel out so that the period of oscillation
doesnrsquot change
Part I
The spring constant of spring 3 can be adjusted with the softness spring 3 slider bar (harder means a greater spring constant or stiffer spring)
How does the frequency of oscillation depend on the spring constant
Hint 1 How to approach the problem
You can place a weight on spring 3 with the softness of spring 3 set to a very low (soft) value and place another weight with the same masson spring 1 Then release the two weights and compare their frequencies of oscillation
ANSWER
Correct
It turns out that the frequency of oscillation depends on the square root of the ratio of the spring constant to mass where
is the frequency A stiffer spring constant causes the frequency to increase Sports cars use stiff springs whereas large plush Cadillacs use softsprings for their suspension
PhET Interactive Simulations
University of Colorado
httpphetcoloradoedu
PhET Tutorial Pendulum Lab
Learning Goal
To understand the relationships of the energetics forces acceleration and velocity of an oscillating pendulum and to determine how the motion of a
pendulum depends on the mass the length of the string and the acceleration due to gravity
The frequency increases as the mass increases
The frequency is independent of the mass
The frequency decreases as the mass increases
The frequency increases as the amplitude increases
The frequency decreases as the amplitude increases
The frequency is independent of the amplitude
The frequency increases as the spring constant increases
The frequency is independent of the spring constant
The frequency decreases as the spring constant increases
For this tutorial use the PhET simulation Pendulum Lab This simulation mimics a real pendulum and allows you to adjust the initial position the mass
and the length of the pendulum
Start the simulation You can drag the pendulum to an arbitrary initial angle and release it from rest You can adjust the length and the mass of the
pendulum using the slider bars at the top of the green panel Velocity and acceleration vectors can be selected to be shown as well as the forms of
energy
Feel free to play around with the simulation When you are done click the Reset button
Part A
Select to show the energy of pendulum 1 Be sure that friction is set to none
Drag the pendulum to an angle (with respect to the vertical) of and then release itWhen the pendulum is at what form(s) of energy does it have
Check all that apply
ANSWER
Correct
The pendulum starts off with no kinetic energy since it is released from rest so it initially only has potential energy When the pendulum is at
it is just as high above the ground as when it started so it must have the same amount of potential energy as it initially had Since thetotal energy is conserved it canrsquot have other forms of energy at (if it did it would have more energy there than it initially had) so it again has
only potential energy
Part B
Drag the pendulum to an angle (with respect to the vertical) of and then release it
Where is the pendulum swinging the fastest
ANSWER
Correct
The pendulum has the least potential energy at this location since it is at the lowest point in the arc (in fact for this simulation the potential
energy reference location is here so it has no potential energy) This means that the kinetic energy is greatest here so the pendulum is moving
Drag the pendulum to an angle (with respect to the vertical) of and then release it Select to show the acceleration vector
With the pendulum swinging back and forth at which locations is the acceleration equal to zero
ANSWER
Correct
The pendulum is moving in a circular path so its velocity is never constant In fact for most locations the acceleration has both a radial
component (the centripetal acceleration which is directed along the rope) and a tangential component (due to the speed changing directed along
the path the only place the tangential acceleration is zero is when the angle is )
Part D
With the pendulum swinging back and forth how does the tension of the rope compare to the force of gravity when the angle is
Hint 1 How to approach the problem
Look at the direction of the acceleration when the angle is (you can slow down or pause the simulation to see this more clearly) and think
about the relationship between the net force acting on the pendulum and the acceleration (Newtonrsquos 2nd law of motion) It would probably help todraw a free-body diagram for the mass
ANSWER
Correct
Since the acceleration of the pendulum is directed up when the angle is the net force must be directed up (Newtonrsquos 2nd law) This meansthat the upward force of tension must be stronger than the downward force of gravity
Part E
Drag the pendulum to an angle (with respect to the vertical) of and then release it
With the pendulum swinging back and forth where is the tension equal to zero
ANSWER
Correct
At these locations the acceleration is solely due to gravity and directed downward Thus the net force acting on t he pendulum i s also directed
downward meaning there are no horizontal forces This requires the tension to be zero
Part F
Now for parts F-I you will investigate how the period of oscillation depends on the properties of the pendulum
The period of oscillation is the amount of time it takes for the pendulum to take a full swing going from the original angle to the other side and
3 0
∘
The acceleration is zero when the angle is either or
The acceleration is zero when the angle is
The acceleration is never equal to zero as it swings back and forth
+ 3 0
∘
minus 3 0
∘
0
∘
0
∘
0
∘
0
∘
The tension is equal to the force of gravity
The tension is greater than the force of gravity only if it is swinging really fast
A greater mass results in a lower frequency and a longer period of oscillation
Part H
The amplitude of oscillation is the maximum distance between the oscillating weight and the equilibrium position Determine the frequency of
oscillation for several different amplitudes by pulling the weight down different amounts
How does the frequency depend on the amplitude of oscillation
ANSWER
CorrectEven though the weight has to travel farther each oscillation if the amplitude is greater the spring on average exerts a stronger force causing a
greater acceleration and a greater average speed The effects of the longer distance and faster speed cancel out so that the period of oscillation
doesnrsquot change
Part I
The spring constant of spring 3 can be adjusted with the softness spring 3 slider bar (harder means a greater spring constant or stiffer spring)
How does the frequency of oscillation depend on the spring constant
Hint 1 How to approach the problem
You can place a weight on spring 3 with the softness of spring 3 set to a very low (soft) value and place another weight with the same masson spring 1 Then release the two weights and compare their frequencies of oscillation
ANSWER
Correct
It turns out that the frequency of oscillation depends on the square root of the ratio of the spring constant to mass where
is the frequency A stiffer spring constant causes the frequency to increase Sports cars use stiff springs whereas large plush Cadillacs use softsprings for their suspension
PhET Interactive Simulations
University of Colorado
httpphetcoloradoedu
PhET Tutorial Pendulum Lab
Learning Goal
To understand the relationships of the energetics forces acceleration and velocity of an oscillating pendulum and to determine how the motion of a
pendulum depends on the mass the length of the string and the acceleration due to gravity
The frequency increases as the mass increases
The frequency is independent of the mass
The frequency decreases as the mass increases
The frequency increases as the amplitude increases
The frequency decreases as the amplitude increases
The frequency is independent of the amplitude
The frequency increases as the spring constant increases
The frequency is independent of the spring constant
The frequency decreases as the spring constant increases
For this tutorial use the PhET simulation Pendulum Lab This simulation mimics a real pendulum and allows you to adjust the initial position the mass
and the length of the pendulum
Start the simulation You can drag the pendulum to an arbitrary initial angle and release it from rest You can adjust the length and the mass of the
pendulum using the slider bars at the top of the green panel Velocity and acceleration vectors can be selected to be shown as well as the forms of
energy
Feel free to play around with the simulation When you are done click the Reset button
Part A
Select to show the energy of pendulum 1 Be sure that friction is set to none
Drag the pendulum to an angle (with respect to the vertical) of and then release itWhen the pendulum is at what form(s) of energy does it have
Check all that apply
ANSWER
Correct
The pendulum starts off with no kinetic energy since it is released from rest so it initially only has potential energy When the pendulum is at
it is just as high above the ground as when it started so it must have the same amount of potential energy as it initially had Since thetotal energy is conserved it canrsquot have other forms of energy at (if it did it would have more energy there than it initially had) so it again has
only potential energy
Part B
Drag the pendulum to an angle (with respect to the vertical) of and then release it
Where is the pendulum swinging the fastest
ANSWER
Correct
The pendulum has the least potential energy at this location since it is at the lowest point in the arc (in fact for this simulation the potential
energy reference location is here so it has no potential energy) This means that the kinetic energy is greatest here so the pendulum is moving
Drag the pendulum to an angle (with respect to the vertical) of and then release it Select to show the acceleration vector
With the pendulum swinging back and forth at which locations is the acceleration equal to zero
ANSWER
Correct
The pendulum is moving in a circular path so its velocity is never constant In fact for most locations the acceleration has both a radial
component (the centripetal acceleration which is directed along the rope) and a tangential component (due to the speed changing directed along
the path the only place the tangential acceleration is zero is when the angle is )
Part D
With the pendulum swinging back and forth how does the tension of the rope compare to the force of gravity when the angle is
Hint 1 How to approach the problem
Look at the direction of the acceleration when the angle is (you can slow down or pause the simulation to see this more clearly) and think
about the relationship between the net force acting on the pendulum and the acceleration (Newtonrsquos 2nd law of motion) It would probably help todraw a free-body diagram for the mass
ANSWER
Correct
Since the acceleration of the pendulum is directed up when the angle is the net force must be directed up (Newtonrsquos 2nd law) This meansthat the upward force of tension must be stronger than the downward force of gravity
Part E
Drag the pendulum to an angle (with respect to the vertical) of and then release it
With the pendulum swinging back and forth where is the tension equal to zero
ANSWER
Correct
At these locations the acceleration is solely due to gravity and directed downward Thus the net force acting on t he pendulum i s also directed
downward meaning there are no horizontal forces This requires the tension to be zero
Part F
Now for parts F-I you will investigate how the period of oscillation depends on the properties of the pendulum
The period of oscillation is the amount of time it takes for the pendulum to take a full swing going from the original angle to the other side and
3 0
∘
The acceleration is zero when the angle is either or
The acceleration is zero when the angle is
The acceleration is never equal to zero as it swings back and forth
+ 3 0
∘
minus 3 0
∘
0
∘
0
∘
0
∘
0
∘
The tension is equal to the force of gravity
The tension is greater than the force of gravity only if it is swinging really fast
A greater mass results in a lower frequency and a longer period of oscillation
Part H
The amplitude of oscillation is the maximum distance between the oscillating weight and the equilibrium position Determine the frequency of
oscillation for several different amplitudes by pulling the weight down different amounts
How does the frequency depend on the amplitude of oscillation
ANSWER
CorrectEven though the weight has to travel farther each oscillation if the amplitude is greater the spring on average exerts a stronger force causing a
greater acceleration and a greater average speed The effects of the longer distance and faster speed cancel out so that the period of oscillation
doesnrsquot change
Part I
The spring constant of spring 3 can be adjusted with the softness spring 3 slider bar (harder means a greater spring constant or stiffer spring)
How does the frequency of oscillation depend on the spring constant
Hint 1 How to approach the problem
You can place a weight on spring 3 with the softness of spring 3 set to a very low (soft) value and place another weight with the same masson spring 1 Then release the two weights and compare their frequencies of oscillation
ANSWER
Correct
It turns out that the frequency of oscillation depends on the square root of the ratio of the spring constant to mass where
is the frequency A stiffer spring constant causes the frequency to increase Sports cars use stiff springs whereas large plush Cadillacs use softsprings for their suspension
PhET Interactive Simulations
University of Colorado
httpphetcoloradoedu
PhET Tutorial Pendulum Lab
Learning Goal
To understand the relationships of the energetics forces acceleration and velocity of an oscillating pendulum and to determine how the motion of a
pendulum depends on the mass the length of the string and the acceleration due to gravity
The frequency increases as the mass increases
The frequency is independent of the mass
The frequency decreases as the mass increases
The frequency increases as the amplitude increases
The frequency decreases as the amplitude increases
The frequency is independent of the amplitude
The frequency increases as the spring constant increases
The frequency is independent of the spring constant
The frequency decreases as the spring constant increases
For this tutorial use the PhET simulation Pendulum Lab This simulation mimics a real pendulum and allows you to adjust the initial position the mass
and the length of the pendulum
Start the simulation You can drag the pendulum to an arbitrary initial angle and release it from rest You can adjust the length and the mass of the
pendulum using the slider bars at the top of the green panel Velocity and acceleration vectors can be selected to be shown as well as the forms of
energy
Feel free to play around with the simulation When you are done click the Reset button
Part A
Select to show the energy of pendulum 1 Be sure that friction is set to none
Drag the pendulum to an angle (with respect to the vertical) of and then release itWhen the pendulum is at what form(s) of energy does it have
Check all that apply
ANSWER
Correct
The pendulum starts off with no kinetic energy since it is released from rest so it initially only has potential energy When the pendulum is at
it is just as high above the ground as when it started so it must have the same amount of potential energy as it initially had Since thetotal energy is conserved it canrsquot have other forms of energy at (if it did it would have more energy there than it initially had) so it again has
only potential energy
Part B
Drag the pendulum to an angle (with respect to the vertical) of and then release it
Where is the pendulum swinging the fastest
ANSWER
Correct
The pendulum has the least potential energy at this location since it is at the lowest point in the arc (in fact for this simulation the potential
energy reference location is here so it has no potential energy) This means that the kinetic energy is greatest here so the pendulum is moving
Drag the pendulum to an angle (with respect to the vertical) of and then release it Select to show the acceleration vector
With the pendulum swinging back and forth at which locations is the acceleration equal to zero
ANSWER
Correct
The pendulum is moving in a circular path so its velocity is never constant In fact for most locations the acceleration has both a radial
component (the centripetal acceleration which is directed along the rope) and a tangential component (due to the speed changing directed along
the path the only place the tangential acceleration is zero is when the angle is )
Part D
With the pendulum swinging back and forth how does the tension of the rope compare to the force of gravity when the angle is
Hint 1 How to approach the problem
Look at the direction of the acceleration when the angle is (you can slow down or pause the simulation to see this more clearly) and think
about the relationship between the net force acting on the pendulum and the acceleration (Newtonrsquos 2nd law of motion) It would probably help todraw a free-body diagram for the mass
ANSWER
Correct
Since the acceleration of the pendulum is directed up when the angle is the net force must be directed up (Newtonrsquos 2nd law) This meansthat the upward force of tension must be stronger than the downward force of gravity
Part E
Drag the pendulum to an angle (with respect to the vertical) of and then release it
With the pendulum swinging back and forth where is the tension equal to zero
ANSWER
Correct
At these locations the acceleration is solely due to gravity and directed downward Thus the net force acting on t he pendulum i s also directed
downward meaning there are no horizontal forces This requires the tension to be zero
Part F
Now for parts F-I you will investigate how the period of oscillation depends on the properties of the pendulum
The period of oscillation is the amount of time it takes for the pendulum to take a full swing going from the original angle to the other side and
3 0
∘
The acceleration is zero when the angle is either or
The acceleration is zero when the angle is
The acceleration is never equal to zero as it swings back and forth
+ 3 0
∘
minus 3 0
∘
0
∘
0
∘
0
∘
0
∘
The tension is equal to the force of gravity
The tension is greater than the force of gravity only if it is swinging really fast
For this tutorial use the PhET simulation Pendulum Lab This simulation mimics a real pendulum and allows you to adjust the initial position the mass
and the length of the pendulum
Start the simulation You can drag the pendulum to an arbitrary initial angle and release it from rest You can adjust the length and the mass of the
pendulum using the slider bars at the top of the green panel Velocity and acceleration vectors can be selected to be shown as well as the forms of
energy
Feel free to play around with the simulation When you are done click the Reset button
Part A
Select to show the energy of pendulum 1 Be sure that friction is set to none
Drag the pendulum to an angle (with respect to the vertical) of and then release itWhen the pendulum is at what form(s) of energy does it have
Check all that apply
ANSWER
Correct
The pendulum starts off with no kinetic energy since it is released from rest so it initially only has potential energy When the pendulum is at
it is just as high above the ground as when it started so it must have the same amount of potential energy as it initially had Since thetotal energy is conserved it canrsquot have other forms of energy at (if it did it would have more energy there than it initially had) so it again has
only potential energy
Part B
Drag the pendulum to an angle (with respect to the vertical) of and then release it
Where is the pendulum swinging the fastest
ANSWER
Correct
The pendulum has the least potential energy at this location since it is at the lowest point in the arc (in fact for this simulation the potential
energy reference location is here so it has no potential energy) This means that the kinetic energy is greatest here so the pendulum is moving
Drag the pendulum to an angle (with respect to the vertical) of and then release it Select to show the acceleration vector
With the pendulum swinging back and forth at which locations is the acceleration equal to zero
ANSWER
Correct
The pendulum is moving in a circular path so its velocity is never constant In fact for most locations the acceleration has both a radial
component (the centripetal acceleration which is directed along the rope) and a tangential component (due to the speed changing directed along
the path the only place the tangential acceleration is zero is when the angle is )
Part D
With the pendulum swinging back and forth how does the tension of the rope compare to the force of gravity when the angle is
Hint 1 How to approach the problem
Look at the direction of the acceleration when the angle is (you can slow down or pause the simulation to see this more clearly) and think
about the relationship between the net force acting on the pendulum and the acceleration (Newtonrsquos 2nd law of motion) It would probably help todraw a free-body diagram for the mass
ANSWER
Correct
Since the acceleration of the pendulum is directed up when the angle is the net force must be directed up (Newtonrsquos 2nd law) This meansthat the upward force of tension must be stronger than the downward force of gravity
Part E
Drag the pendulum to an angle (with respect to the vertical) of and then release it
With the pendulum swinging back and forth where is the tension equal to zero
ANSWER
Correct
At these locations the acceleration is solely due to gravity and directed downward Thus the net force acting on t he pendulum i s also directed
downward meaning there are no horizontal forces This requires the tension to be zero
Part F
Now for parts F-I you will investigate how the period of oscillation depends on the properties of the pendulum
The period of oscillation is the amount of time it takes for the pendulum to take a full swing going from the original angle to the other side and
3 0
∘
The acceleration is zero when the angle is either or
The acceleration is zero when the angle is
The acceleration is never equal to zero as it swings back and forth
+ 3 0
∘
minus 3 0
∘
0
∘
0
∘
0
∘
0
∘
The tension is equal to the force of gravity
The tension is greater than the force of gravity only if it is swinging really fast
Drag the pendulum to an angle (with respect to the vertical) of and then release it Select to show the acceleration vector
With the pendulum swinging back and forth at which locations is the acceleration equal to zero
ANSWER
Correct
The pendulum is moving in a circular path so its velocity is never constant In fact for most locations the acceleration has both a radial
component (the centripetal acceleration which is directed along the rope) and a tangential component (due to the speed changing directed along
the path the only place the tangential acceleration is zero is when the angle is )
Part D
With the pendulum swinging back and forth how does the tension of the rope compare to the force of gravity when the angle is
Hint 1 How to approach the problem
Look at the direction of the acceleration when the angle is (you can slow down or pause the simulation to see this more clearly) and think
about the relationship between the net force acting on the pendulum and the acceleration (Newtonrsquos 2nd law of motion) It would probably help todraw a free-body diagram for the mass
ANSWER
Correct
Since the acceleration of the pendulum is directed up when the angle is the net force must be directed up (Newtonrsquos 2nd law) This meansthat the upward force of tension must be stronger than the downward force of gravity
Part E
Drag the pendulum to an angle (with respect to the vertical) of and then release it
With the pendulum swinging back and forth where is the tension equal to zero
ANSWER
Correct
At these locations the acceleration is solely due to gravity and directed downward Thus the net force acting on t he pendulum i s also directed
downward meaning there are no horizontal forces This requires the tension to be zero
Part F
Now for parts F-I you will investigate how the period of oscillation depends on the properties of the pendulum
The period of oscillation is the amount of time it takes for the pendulum to take a full swing going from the original angle to the other side and
3 0
∘
The acceleration is zero when the angle is either or
The acceleration is zero when the angle is
The acceleration is never equal to zero as it swings back and forth
+ 3 0
∘
minus 3 0
∘
0
∘
0
∘
0
∘
0
∘
The tension is equal to the force of gravity
The tension is greater than the force of gravity only if it is swinging really fast