# AP Physics 1 : Pendulums

## Example Questions

### Example Question #1 : Pendulums

A ball of mass 2kg is attached to a string of length 4m, forming a pendulum. If the string is raised to have an angle of 30 degrees below the horizontal and released, what is the velocity of the ball as it passes through its lowest point?

Explanation:

This question deals with conservation of energy in the form of a pendulum. The equation for conservation of energy is:

According to the problem statement, there is no initial kinetic energy and no final potential energy. The equation becomes:

Subsituting in the expressions for potential and kinetic energy, we get:

We can eliminate mass to get:

Rearranging for final velocity, we get:

In order to solve for the velocity, we need to find the initial height of the ball.

The following diagram will help visualize the system:

From this, we can write:

Using the length of string and the angle it's held at, we can solve for :

Now that we have all of our information, we can solve for the final velocity:

### Example Question #2 : Pendulums

A pendulum has a period of 5 seconds. If the length of the string of the pendulum is quadrupled, what is the new period of the pendulum?

Explanation:

We need to know how to calculate the period of a pendulum to solve this problem. The formula for period is:

In the problem, we are only changing the length of the string. Therefore, we can rewrite the equation for each scenario:

Dividng one expression by the other, we get a ratio:

We know that , so we can rewrite the expression as:

Rearranging for P2, we get:

### Example Question #2 : Pendulums

A student studying Newtonian mechanics in the 19th century was skeptical of some of Newton's concepts. The student has a pendulum that has a period of 3 seconds while sitting on his desk. He attaches the pendulum to a ballon and drops it off the roof of a university building, which is 20m tall. Another student realizes that the pendulum strikes the ground with a velocity of . What is the period of the pendulum as it is falling to the ground?

Neglect air resistance and assume

Explanation:

We need to know the formula for the period of a pendulum to solve this problem:

We aren't given the length of the pendulum, but that's ok. We could solve for it, but it would be an unnecessary step since the length remains constant.

We can write this formula for the pendulum when it is on the student's table and when it is falling:

1 denotes on the table and 2 denotes falling. The only thing that is different between the two states is the period and the gravity (technically the acceleration of the whole system, but this is the form in which you are most likely to see the formula). We can divide the two expressions to get a ratio:

Canceling out the constants and rearranging, we get:

We know g1; it's simply 10. However, we need to calculate g2, which is the rate at which the pendulum and balloon are accelerating toward the ground. We are given enough information to use the following formula to determine this:

Removing initial velocity and rearranging for acceleration, we get:

Plugging in our values:

This is our g2. We now have all of the values to solve for T2:

### Example Question #1 : Pendulums

A pendulum of mass  has a period . If the mass is quadrupled to , what is the new period of the pendulum in terms of ?

Explanation:

The mass of a pendulum has no effect on its period. The equation for the period of a pendulum is

, which does not include mass.

### Example Question #1 : Pendulums

A pendulum of length  will take how long to complete one period of its swing?

Explanation:

The period of a pendulum is given by the following formula:

Substituting our values, we obtain:

Roughly 6.3 seconds is the time it takes for the pendulum to complete one period.

### Example Question #6 : Pendulums

In the lab, a student has created a pendulum by hanging a weight from a string. The student releases the pendulum from rest and uses a sensor and computer to find the equation of motion for the pendulum:

The student then replaces the weight with a weight whose mass,  is twice as large as that of the original weight without changing the length of the string. The student again releases the weight from rest from the same displacement from equilibrium. What would the new equation of motion be for the pendulum?

Explanation:

The period and frequency of a pendulum depend only on its length and the gravity force constant, .  Changing the mass of the pendulum does not affect the frequency, and since the student released the new pendulum from the same displacement as the old, the amplitude and phase remain the same, and the equation of motion is the same for both pendula.

### Example Question #7 : Pendulums

In the lab, a student has created a pendulum by hanging a weight from a string. The student releases the pendulum from rest and uses a sensor and computer to find the equation of motion for the pendulum:

The student then replaces the string with a string whose length,  is twice as large as that of the original string without changing the mass of the weight. The student again releases the weight from rest from the same displacement from equilibrium. What would the new equation of motion be for the pendulum?

Explanation:

Doubling the length of a pendulum increases the period, so it decreases the frequency of the pendulum. The frequency depends upon the square root of the length, so the frequency decreases by a factor of . Neither of the other parameters (amplitude, phase) change.

### Example Question #8 : Pendulums

A pendulum of length has a mass of attached to the bottom. Determine the frequency of the pendulum if it is released from a shallow angle.

Explanation:

The frequency of a pendulum is given by:

Where is the length of the pendulum and is the gravity constant. Notice how the frequency is independent of mass.

Plugging in values:

### Example Question #1 : Pendulums

How will increasing the mass at the end of a pendulum change the period of it's motion? Assume a shallow angle of release.

It depends on how much mass is added

It will increase

It will decrease

There will be no change

There will be no change

Explanation:

The frequency of a pendulum is given by:

Where is the length of the pendulum and is the gravity constant. The frequency is independent of mass. Thus, adding mass will have no effect.

### Example Question #1 : Pendulums

If a simple pendulum is set to oscillate on Earth, it has a period of . Now suppose this same pendulum were moved to the Moon, where the gravitational field is 6 times less than that of Earth.

What is the period  of this pendulum on the Moon in terms of ?

Explanation:

The period of a simple pendulum is given by:

Where  is the period of the pendulum,  is the length of the pendulum, and  is the gravitational constant of the planet we are on. Thus on Earth, the period  is given by:

With  being Earth's gravitational constant. The period on the Moon is given by:

With  being the Moon's gravitational constant. Since the Moon's gravity is 6 times weaker than that of Earth's, we have:

Plug this value into the Moon pendulum equation:

Since ,

Substituting this into the above expression gives us