# AP Physics 1 : Period and Frequency of Harmonic Motion

## Example Questions

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### Example Question #1 : Period And Frequency Of Harmonic Motion

A horizontal spring on a frictionless surface has a spring constant of  with a mass of  attached to the end of the spring. If the spring is stretched  passed its point of equilibrium and released, how many times does the mass pass through equilibrium per minute?

Explanation:

We need to know three things to solve this problem.

1. Formula for frequency of a spring in simple harmonic motion
2. The spring constant
3. Mass of object attached to spring

We're given 2 and 3, so we just need to know 1.

The formula for frequency is:

Plugging in our values, we get:

Now we need to convert Hertz (cycles per second) to cycles per minute.

To get to these units:

### Example Question #2 : Period And Frequency Of Harmonic Motion

Consider the following system:

If the period of the pendulum is , what is the length of the string?

Explanation:

We simply need the expression for the period of a pendulum to solve this problem:

Rearranging for length, we get:

We have all of these values, allowing us to solve:

### Example Question #3 : Period And Frequency Of Harmonic Motion

Consider the following system:

If the mass reaches a maximum height of  and the minimum angle is , what is the period of the pendulum?

Explanation:

First, we will be using the equation for the period of a pendulum:

The only value we don't have is length. However, we can develop an expression for length from the given information.

The second term describes how close the pendulum gets to the height of the top of the pendulum. Therefore, we subtract this value from the lowest point of the pendulum to get the height relative to its lowest point. We can rewrite this as:

Substituting this into the original expression for the period, we get:

We can use our given values to solve:

### Example Question #4 : Period And Frequency Of Harmonic Motion

A block of mass 1.3 kg is attached to a spring whose force constant, , is .  What is the frequency and period of the oscillations of this spring-block system?

Explanation:

For a mass-spring system undergoing simple harmonic motion, the frequency of the oscillations can be found using the equation

We were given the force constant (or spring constant), , to be . The oscillating mass was also given to be 1.3 kg. So, plug these in to the equation and solve for frequency, . The unit for frequency is Hertz, Hz.

### Example Question #5 : Period And Frequency Of Harmonic Motion

A 300 g mass is attached to a spring and undergoes simple harmonic motion with a period of 0.25 s.  If the total energy of the system is 3.0 J, what is the aplitude of the oscillations?

Explanation:

We were given the mass of the system as 300 g. First, we should convert this to kilograms. Since , we can convert by

They also told us the period of the oscillations, which is 0.25 s.  We can use the following equation to solve for the force constant:

Rearrange the equation to isolate the variable , and substitute in the known values to solve for :

Now that we have the force constant, , we can use this to help us find the amplitude of the oscillations. We were also given the energy of the system to be 3.0 J. The equation for the energy of a mass-spring oscillating system is:

is the total potential energy of the system. Now, plug in values and solve for , which is the amplitude of the oscillations.

### Example Question #6 : Period And Frequency Of Harmonic Motion

A pendulum of length  undergoing simple harmonic motion has a period of 1.0 s on Earth. If the acceleration of gravity on the Moon is , what would the period of the pendulum be on the Moon?

Explanation:

The equation that relates the period of a pendulum and its length is

We know the period of the pendulum on Earth, which was given, to be 1.0 s. We also know that  on Earth is .  So, we can find the length of the pendulum:

Now that we have the length of the pendulum, we can use this to calculate what the period of oscillation will be if the pendulum was on the Moon where the "g" is not the same as on Earth, but rather is .

### Example Question #7 : Period And Frequency Of Harmonic Motion

If a period of a pendulum is , what is its length?

None of these

Explanation:

Equation for period of a pendulum:

Solve.

### Example Question #8 : Period And Frequency Of Harmonic Motion

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

The student then replaces the weight with a weight whose mass,  is twice as large as that of the original 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 oscillator?

Explanation:

The equation of motion for an oscillator is:

In words, the position as a function of time equals the amplitude times the cosine of the sum of the frequency (in radians per second) times time plus phase. Changing the mass of the weight only changes the frequency, but it depends upon the square root of the mass. Since increasing the mass increases the period, it decreases the frequency by the factor of .

### Example Question #9 : Period And Frequency Of Harmonic Motion

The Fourier Transform is an extensively used mathematical analysis technique. In some applications, it reconstructs a function  into an infinite series of sine waves.

Given that the Fourier Transform is given by the series:

Where  is an arbitrary length and  is an integer, what is the wavelength of the sinusoid when

Explanation:

First, we want to ignore the summation sign and the  since those terms do not affect the wavelength at . All we need to look at is the sine term.

For , we get the sinusoid:

Remember that wavelength  is given by

### Example Question #10 : Period And Frequency Of Harmonic Motion

Determine the period of a sine wave that has a frequency of .