Harmonic Motion - High School Physics
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A bullet with mass 
hits a ballistic pendulum with length 
and mass 
and lodges in it. When the bullet hits the pendulum it swings up from the equilibrium position and reaches an angle 
at its maximum. Determine the bullet’s velocity.
A bullet with mass 
We will need to start at the end of the situation and work backwards in order to determine the velocity of the bullet. At the very end, the pendulum with the bullet reaches its maximum height and therefore comes to a stop. It has gravitational potential energy. At the bottom of the pendulum right after the bullet collides with it, it has kinetic energy due to the velocity of the bullet. With the law of conservation of energy we can set the kinetic energy of the pendulum right after the collision equal to the gravitational potential energy of the pendulum at the highest point.
To determine the height of the pendulum we will need to use trig and triangles to find the height. We know that the pendulum makes a 30 degree angle with the equilibrium position at its maximum height. The length of the pendulum is provided which is the hypotenuse of this triangle. We need to find the adjacent side of this triangle. We can use cosine to determine this.
We can now subtract this value from the length of the pendulum to determine how high off the ground the pendulum is at its highest point.
We can now set the kinetic energy of the pendulum right after the collision equal to the gravitational potential energy of the pendulum at the highest point.
The mass is the same throughout so it falls out of the equation.
The pendulum with the bullet was moving 
after the collision. We can now use momentum to determine the speed of the bullet before the collision. Conservation of momentum states that the momentum before the collision must equal the momentum after the collision.
They both move together after the collision
Since the pendulum was not moving at the beginning
We can now plug in these values and solve for the missing piece.
We will need to start at the end of the situation and work backwards in order to determine the velocity of the bullet. At the very end, the pendulum with the bullet reaches its maximum height and therefore comes to a stop. It has gravitational potential energy. At the bottom of the pendulum right after the bullet collides with it, it has kinetic energy due to the velocity of the bullet. With the law of conservation of energy we can set the kinetic energy of the pendulum right after the collision equal to the gravitational potential energy of the pendulum at the highest point.
To determine the height of the pendulum we will need to use trig and triangles to find the height. We know that the pendulum makes a 30 degree angle with the equilibrium position at its maximum height. The length of the pendulum is provided which is the hypotenuse of this triangle. We need to find the adjacent side of this triangle. We can use cosine to determine this.
We can now subtract this value from the length of the pendulum to determine how high off the ground the pendulum is at its highest point.
We can now set the kinetic energy of the pendulum right after the collision equal to the gravitational potential energy of the pendulum at the highest point.
The mass is the same throughout so it falls out of the equation.
The pendulum with the bullet was moving 
They both move together after the collision
Since the pendulum was not moving at the beginning
We can now plug in these values and solve for the missing piece.
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A spring with a spring constant of 
is compressed 
. How much potential energy has been generated?
A spring with a spring constant of 
The formula for the potential energy in a spring is:
Use the given spring constant and displacement to solve for the stored energy.
The formula for the potential energy in a spring is:
Use the given spring constant and displacement to solve for the stored energy.
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A vertical spring with a spring constant of 
is stationary. An 
mass is attached to the end of the spring. What is the maximum displacement that the spring will stretch?
A vertical spring with a spring constant of 
The best way to solve this problem is by using energy. Notice that the spring on its own is stationary. That means its initial total energy at that moment is zero. When the mass is attached, the spring stretches out, giving it spring potential energy (PEspring).
Where does that energy come from? The only place it can come from is the addition of the mass. Since the system is vertical, this mass will have gravitational potential energy.
Use the law of conservation of energy to set these two energies equal to each other:
We are trying to solve for displacement, and now we have an equation in terms of our variable.
Start by diving both sides by Δy to get rid of the Δy2 on the right side of the equation.
We are given values for the spring constant, the mass, and gravity. Using these values will allow use to solve for the displacement.
Note that the displacement will be negative because the spring is stretched in the downward direction due to gravity.
The best way to solve this problem is by using energy. Notice that the spring on its own is stationary. That means its initial total energy at that moment is zero. When the mass is attached, the spring stretches out, giving it spring potential energy (PEspring).
Where does that energy come from? The only place it can come from is the addition of the mass. Since the system is vertical, this mass will have gravitational potential energy.
Use the law of conservation of energy to set these two energies equal to each other:
We are trying to solve for displacement, and now we have an equation in terms of our variable.
Start by diving both sides by Δy to get rid of the Δy2 on the right side of the equation.
We are given values for the spring constant, the mass, and gravity. Using these values will allow use to solve for the displacement.
Note that the displacement will be negative because the spring is stretched in the downward direction due to gravity.
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A 
mass is placed at the end of a spring. The spring is compressed 
. What is the maximum velocity of the mass if the spring has a spring constant of 
?
A 
If we're looking for the maximum velocity, that will happen when all the energy in the system is kinetic energy.
We can use the law of conservation of energy to see 
. So, if we can find the initial potential energy, we can find the final kinetic energy, and use that to find the mass's final velocity.
The formula for spring potential energy is:
Plug in our given values and solve:
The formula for kinetic energy is:
Since 
, that means that 
.
We can plug in that information to the formula for kinetic energy to solve for the maximum velocity:
If we're looking for the maximum velocity, that will happen when all the energy in the system is kinetic energy.
We can use the law of conservation of energy to see 
The formula for spring potential energy is:
Plug in our given values and solve:
The formula for kinetic energy is:
Since 
We can plug in that information to the formula for kinetic energy to solve for the maximum velocity:
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A pendulum is dropped from rest at 
above the ground. If no outside forces (except gravity) are acting upon it, what will be its maximum height on the other side?
A pendulum is dropped from rest at 
If no outside forces act upon the pendulum, it will continue to oscillate back to the original height of 
.
The proof of this is in the law of conservation of energy. At the top, the pendulum has all potential energy, which is given by the formula 
. As it swings, the potential energy is converted to kinetic energy until, at the bottommost point, there is only kinetic energy. It then changes direction and begins to rise again. When it rises to the maximum height on the other side, all of its kinetic energy will turn back into potential energy.
Mathematically, the initial and final potential energies are equal.
Notice the masses and gravity can cancel out on both sides, as neither of these will change. This leaves us with only height.
If no outside forces act upon the pendulum, it will continue to oscillate back to the original height of 
The proof of this is in the law of conservation of energy. At the top, the pendulum has all potential energy, which is given by the formula 
Mathematically, the initial and final potential energies are equal.
Notice the masses and gravity can cancel out on both sides, as neither of these will change. This leaves us with only height.
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A mass on a string is released and swings freely. Which of the following best explains the energy of the pendulum when the string is perpendicular to the ground?
A mass on a string is released and swings freely. Which of the following best explains the energy of the pendulum when the string is perpendicular to the ground?
Conservation of energy dictates that the total mechanical energy will remain constant. Initially, the mass will not be moving and will be at its highest height. When released, it will begin to travel downward (lose potential energy) and gain velocity (gain kinetic energy). When the mass reaches the bottommost point in the swing, the potential energy will be at a minimum and the kinetic energy will be at a maximum. This point corresponds to the string being perpendicular to the ground.
Conservation of energy dictates that the total mechanical energy will remain constant. Initially, the mass will not be moving and will be at its highest height. When released, it will begin to travel downward (lose potential energy) and gain velocity (gain kinetic energy). When the mass reaches the bottommost point in the swing, the potential energy will be at a minimum and the kinetic energy will be at a maximum. This point corresponds to the string being perpendicular to the ground.
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Derive a formula for the maximum speed of a simple pendulum bob in terms of 
, the length L and the angle of swing 
.
Derive a formula for the maximum speed of a simple pendulum bob in terms of 
We can start out by examining the energy in the pendulum. At the top of the swing, there is gravitational potential energy. At the bottom of the swing there is kinetic energy. The law of conservation of energy states that these two values must be the same.
We know the equations for gravitational potential energy and kinetic energy to be
We can set these equal to each other.
Since the mass is constant, it falls out of both sides of the equation.
Let’s rearrange and solve for v by itself.
We don’t know the height of the pendulum at this point and need to get it in terms of the length of the pendulum.
If we knew the angle that the pendulum made with the vertical equilibrium point, we could determine how far off the ground the pendulum was. We can create a triangle with the hypotenuse as the length of the string and the angle between the pendulum and the equilibrium point.
This adjacent side can then be subtracted from the original length of the pendulum to determine the height off the ground.
We can substitute this back into our equation
We can start out by examining the energy in the pendulum. At the top of the swing, there is gravitational potential energy. At the bottom of the swing there is kinetic energy. The law of conservation of energy states that these two values must be the same.
We know the equations for gravitational potential energy and kinetic energy to be
We can set these equal to each other.
Since the mass is constant, it falls out of both sides of the equation.
Let’s rearrange and solve for v by itself.
We don’t know the height of the pendulum at this point and need to get it in terms of the length of the pendulum.
If we knew the angle that the pendulum made with the vertical equilibrium point, we could determine how far off the ground the pendulum was. We can create a triangle with the hypotenuse as the length of the string and the angle between the pendulum and the equilibrium point.
This adjacent side can then be subtracted from the original length of the pendulum to determine the height off the ground.
We can substitute this back into our equation
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Your grandfather clock’s pendulum has a length of 
. If the clock loses half a minute per day, how should you adjust the length of the pendulum?
Your grandfather clock’s pendulum has a length of 
We also can calculate the total number of seconds in a day.
There are 
seconds in one day.
Therefore we want our clock to swing a certain number of times with a period of 
to equal 
.
We know that our current clock has a certain number of swings with a period of 
to equal 
So we have
We can calculate the current period of the pendulum using the equation
We can set up a ratio of each of these two periods to determine the missing length.
Notice that 2, pi and g are all in both the numerator and denominator and therefore fall out of the problem.
We can now solve for our missing piece.
Square both sides to get rid of the square root.
We should lengthen the pendulum by 
We also can calculate the total number of seconds in a day.
There are 
Therefore we want our clock to swing a certain number of times with a period of 
We know that our current clock has a certain number of swings with a period of 
So we have
We can calculate the current period of the pendulum using the equation
We can set up a ratio of each of these two periods to determine the missing length.
Notice that 2, pi and g are all in both the numerator and denominator and therefore fall out of the problem.
We can now solve for our missing piece.
Square both sides to get rid of the square root.
We should lengthen the pendulum by
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A spring has a 
mass attached to one, which oscillates with a period of 
. What is the frequency?
A spring has a 
The mass has no bearing on the relationship between frequency and period. This relationship is given by the equation:
Given the period, the frequency will be equal to its reciprocal.
The mass has no bearing on the relationship between frequency and period. This relationship is given by the equation:
Given the period, the frequency will be equal to its reciprocal.
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A spring has a spring constant of 
. If a force of 
is used to stretch out the spring, what is the total displacement of the spring?
A spring has a spring constant of 
For this problem, use Hooke's law:
In this formula, 
is the spring constant, 
is the compression of the spring, and 
is the necessary force. We are given the spring constant and the force, allowing us to solve for the displacement.
Plug in our given values and solve.
Note that both the force and the displacement are positive because the stretching force will pull in the positive direction. If the spring were compressed, the change in distance would have been negative.
For this problem, use Hooke's law:
In this formula, 
Plug in our given values and solve.
Note that both the force and the displacement are positive because the stretching force will pull in the positive direction. If the spring were compressed, the change in distance would have been negative.
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A mass is placed at the end of a spring. It has a starting velocity of 
and is allowed to oscillate freely. If the mass has a starting velocity of 
, what would the period be?
A mass is placed at the end of a spring. It has a starting velocity of 
When it comes to the period of a spring, the velocity of the object has no effect.
The equation is 
When it comes to the period of a spring, the velocity of the object has no effect.
The equation is
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A spring is attached to a mass, oscillating freely in simple harmonic motion. What change can be made to increase the period of the oscillation?
A spring is attached to a mass, oscillating freely in simple harmonic motion. What change can be made to increase the period of the oscillation?
The equation for the period of a spring in simple harmonic motion is:
In this formula, m is the mass and k is the spring constant. The only two things we can adjust that can change the period, then, are the mass and spring constant. The length of the spring and the acceleration due to gravity are irrelevant.
If we increase the mass we get a larger numerator, which in turns will give us a larger period. If we decrease the mass we get a smaller numerator, which would give us a smaller period. If we use a higher spring constant we get a larger denominator, which also gives us a smaller period.
The equation for the period of a spring in simple harmonic motion is:
In this formula, m is the mass and k is the spring constant. The only two things we can adjust that can change the period, then, are the mass and spring constant. The length of the spring and the acceleration due to gravity are irrelevant.
If we increase the mass we get a larger numerator, which in turns will give us a larger period. If we decrease the mass we get a smaller numerator, which would give us a smaller period. If we use a higher spring constant we get a larger denominator, which also gives us a smaller period.
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A man swings a bucket back and forth at the end of a rope, creating a pendulum. What factor could be used to increase the period of the pendulum?
A man swings a bucket back and forth at the end of a rope, creating a pendulum. What factor could be used to increase the period of the pendulum?
The equation for the period of a pendulum is:
Notice that the material of the pendulum and the mass at the end do not enter into the equation at all. The only things that are capable of affecting the period are the length of the pendulum and the acceleration due to gravity.
While changing the altitude of the pendulum will change the period, it will do it only slightly unless you take it miles above the earth's crust. Furthermore, decreasing the altitude of the pendulum will increase the force due to gravity, which will result in a decreased period. The best answer is to increase the length of the rope. This will increase the period of the pendulum.
The equation for the period of a pendulum is:
Notice that the material of the pendulum and the mass at the end do not enter into the equation at all. The only things that are capable of affecting the period are the length of the pendulum and the acceleration due to gravity.
While changing the altitude of the pendulum will change the period, it will do it only slightly unless you take it miles above the earth's crust. Furthermore, decreasing the altitude of the pendulum will increase the force due to gravity, which will result in a decreased period. The best answer is to increase the length of the rope. This will increase the period of the pendulum.
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What is the period of a simple pendulum 
long when it is in a freely falling elevator?
What is the period of a simple pendulum 
The period of a simple pendulum on the free-falling is infinite because at a time of free-falling elevator the gravitational constant will equal zero which leads to an increase in the value in the numerator to infinite. Thus, the period of time in a free-falling elevator is infinite.
The period of a simple pendulum on the free-falling is infinite because at a time of free-falling elevator the gravitational constant will equal zero which leads to an increase in the value in the numerator to infinite. Thus, the period of time in a free-falling elevator is infinite.
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If a particle undergoes Simple Harmonic motion with an amplitude of 
. What is the total distance it travels in one period?
If a particle undergoes Simple Harmonic motion with an amplitude of 
The period is how long it takes to complete one oscillation. In this case, a complete oscillation would take the object 
away from the center (the amplitude), travel back to the center, travel below the equilibrium point 
away, and then travel back to the center. Therefore the total distance traveled is 
times the amplitude as it must travel away, back, below, and back again. Therefore the total distance traveled is 
.
The period is how long it takes to complete one oscillation. In this case, a complete oscillation would take the object 
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How long must a simple pendulum be if it is to make exactly one swing per second? (That is one complete oscillation would take 2 seconds.)
How long must a simple pendulum be if it is to make exactly one swing per second? (That is one complete oscillation would take 2 seconds.)
The equation to determine the period of pendulum is
We can rearrange this equation to solve for length (
).
We can now solve for the length knowing that the period of the pendulum is 2 seconds.
The equation to determine the period of pendulum is
We can rearrange this equation to solve for length (
We can now solve for the length knowing that the period of the pendulum is 2 seconds.
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A pendulum has a period of 
on Earth. What is its period on Mars, where the acceleration due to gravity is about 
that on Earth?
A pendulum has a period of 
To begin we want to determine the original length of the pendulum.
The equation to determine the period of pendulum is
We can rearrange this equation to solve for length (
).
We can now solve for the length knowing that the period of the pendulum is 2 seconds.
Now we can use the same equation and instead substitute in the value for the acceleration due to gravity on Mars.
To begin we want to determine the original length of the pendulum.
The equation to determine the period of pendulum is
We can rearrange this equation to solve for length (
We can now solve for the length knowing that the period of the pendulum is 2 seconds.
Now we can use the same equation and instead substitute in the value for the acceleration due to gravity on Mars.
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A pendulum is dropped from rest at 
above the ground. If no outside forces (except gravity) are acting upon it, what will be its maximum height on the other side?
A pendulum is dropped from rest at
If no outside forces act upon the pendulum, it will continue to oscillate back to the original height of 
.
The proof of this is in the law of conservation of energy. At the top, the pendulum has all potential energy, which is given by the formula 
. As it swings, the potential energy is converted to kinetic energy until, at the bottommost point, there is only kinetic energy. It then changes direction and begins to rise again. When it rises to the maximum height on the other side, all of its kinetic energy will turn back into potential energy.
Mathematically, the initial and final potential energies are equal.
Notice the masses and gravity can cancel out on both sides, as neither of these will change. This leaves us with only height.
If no outside forces act upon the pendulum, it will continue to oscillate back to the original height of
The proof of this is in the law of conservation of energy. At the top, the pendulum has all potential energy, which is given by the formula
Mathematically, the initial and final potential energies are equal.
Notice the masses and gravity can cancel out on both sides, as neither of these will change. This leaves us with only height.
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A mass on a string is released and swings freely. Which of the following best explains the energy of the pendulum when the string is perpendicular to the ground?
A mass on a string is released and swings freely. Which of the following best explains the energy of the pendulum when the string is perpendicular to the ground?
Conservation of energy dictates that the total mechanical energy will remain constant. Initially, the mass will not be moving and will be at its highest height. When released, it will begin to travel downward (lose potential energy) and gain velocity (gain kinetic energy). When the mass reaches the bottommost point in the swing, the potential energy will be at a minimum and the kinetic energy will be at a maximum. This point corresponds to the string being perpendicular to the ground.
Conservation of energy dictates that the total mechanical energy will remain constant. Initially, the mass will not be moving and will be at its highest height. When released, it will begin to travel downward (lose potential energy) and gain velocity (gain kinetic energy). When the mass reaches the bottommost point in the swing, the potential energy will be at a minimum and the kinetic energy will be at a maximum. This point corresponds to the string being perpendicular to the ground.
Compare your answer with the correct one above
A man swings a bucket back and forth at the end of a rope, creating a pendulum. What factor could be used to increase the period of the pendulum?
A man swings a bucket back and forth at the end of a rope, creating a pendulum. What factor could be used to increase the period of the pendulum?
The equation for the period of a pendulum is:
Notice that the material of the pendulum and the mass at the end do not enter into the equation at all. The only things that are capable of affecting the period are are the length of the pendulum and the acceleration due to gravity.
While changing the altitude of the pendulum will change the period, it will do it only slightly unless you take it miles above the earth's crust. Furthermore, decreasing the altitude of the pendulum will increase the force due to gravity, which will resulut in a decreased period. The best answer is to increase the length of the rope. This will increase the period of the pendulum.
The equation for the period of a pendulum is:
Notice that the material of the pendulum and the mass at the end do not enter into the equation at all. The only things that are capable of affecting the period are are the length of the pendulum and the acceleration due to gravity.
While changing the altitude of the pendulum will change the period, it will do it only slightly unless you take it miles above the earth's crust. Furthermore, decreasing the altitude of the pendulum will increase the force due to gravity, which will resulut in a decreased period. The best answer is to increase the length of the rope. This will increase the period of the pendulum.
Compare your answer with the correct one above