Physics 208 – Introductory Physics II

Nazareth College

Department of Chemistry & Biochemistry

Robert F Szalapski, PhD – Adjunct Lecturer

www.CallMeDrRob.com

Spring 2012

Exam 2

Oscillations & Waves

## § Problem 1

The motion of a mass hanging on a spring with spring constant is given by the following equation:

### § (a)

What is the amplitude, , of the motion, and what is the total distance traveled during the first cycle? (A simple sketch may be useful.)

(5 points)

Compare with the form

hence

In one full cycle it travels from through to , then back through to . Hence the total distance traveled is .

### § (b)

What is the frequency, , of the oscillations?

(5 points)

Compare with the form

hence

### § (c)

What is the value of the spring constant, ?

(5 points)

### § (d)

What is the maximum speed, , of the mass?

(5 points)

### § (e)

What is the maximum acceleration, , of the mass?

(5 points)

### § (f)

At what time does the mass first reach the position ?

(5 points)

WARNING: Your calculator should be in radian mode! Otherwise you need to convert from degrees to radians for a physically meaningful result.

### § (g)

What is the position of the mass at ?

(5 points)

WARNING: Your calculator should be in radian mode!

There is another way to see this solution. Above we calculated the frequency, . From this we can get the period, .

If we observe that

we see that the mass has undergone 20 complete cycles, hence it should be exactly where it was at time .

## § Problem 2

Working with a guitar string in the lab a student finds a harmonic with many nodes at . The next harmonic occurs at . The string has a total length of , but the portion that vibrates between the first end to where it passes over a pulley is . The tension is created by hanging a mass from the second end.

### § (a)

What is the fundamental frequency for this string? (10 points)

Recall that

If we consider two adjacent harmonics, then

All we must do to find is to difference two adjacent frequencies.

### § (b)

Which two harmonics were observed by the student in the lab? (10 points)

Recall that

The student observed the and harmonics.

### § (c)

What is the mass of the wire? (10 points)

First let’s find the wave speed using

noting that the wavelength for the fundamental is twice the length of the vibrating portion of the string. So

Next we note that the wave velocity may be determined from the string length, mass and tension as

## § Problem 3

As shown in the picture, a block with unknown mass is attached to a spring with an unknown spring constant all placed on a frictionless table.

### § (a)

A second block is attached to the first by a string and allowed to hang off the end of the table with the string routed over a frictionless pulley causing the spring to stretch by . What is the spring constant ? (10 points)

The Hooke’s Law spring force will be equal to the weight of the suspended block. (Transmitted through the string tension.)

### § (b)

The string is cut, and the block begins to oscillate with a frequency of . What is the mass of the block? (5 points)

### § (c)

What is the maximum speed, , obtained by the block? (10 points)

## § Problem 4

Planet | g |
---|---|

Mercury | |

Venus | |

Earth | |

Mars | |

Jupiter | |

Saturn | |

Uranus | |

Neptune | |

Pluto |

A pendulum undergoes approximately 17 complete oscillations in one full minute. According to Table 1, where in the solar system is the pendulum most likely located? (No credit for guessing without a sensible calculation.)

We know that

We may obtain the period from the information given.

and hence

The answer, within the one significant figure obtained, is Pluto.