Textbook Guide Physics: Kinematics, Moment Of Inertia, Ob River

Good Vibes: Introduction to Oscillations

The conditions that lead to simple harmonic motion are as follows:

There must be a position of stable equilibrium.

There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗  is given by ⃗, where is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.

The resistive forces in the system must be reasonably small.

Consider a block of mass m attached to a spring with force constant k, as shown in the figure(Figure 1) . The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.

Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.

Part A

After the block is released from x=A, it will

(a) remain at rest

(b) move to the left until it reaches equilibrium and stop there. 

(c) move to the left until it reaches x= –A and stop there. 

(d) move to the left until it reaches x=−A and then begin to move to the right.

The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds.

The frequency, denoted f, is the number of cycles that are completed per unit of time: f=1/T. In SI units, f is measured in inverse seconds, or hertz (Hz).

Part B

If the period is doubled, the frequency is

(a) unchanged 

(b) doubled 

(c) halved

Part C

An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f?

Express your answer in hertz.

Part D

If the frequency is 40 Hz, what is the period T?

Express your answer in seconds.

The following questions refer to the figure (Figure 2) that graphically depicts the oscillations of the block on the spring.

Note that the vertical axis represents the x-coordinate of the oscillating object, and the horizontal axis represents time.

Part E

Which points on the x-axis are located at a distance A from the equilibrium position?

(a) R only

(b) Q only

(c) both R and Q

Part F

Suppose that the period is T. Which of the following points on the t-axis are separated by the time interval T?

(a) K and L

(b) K and M 

(c) K and P

(d) L and N 

(e) M and P 

Now assume for the remaining Parts G - J, that the x coordinate of point R is 0.12 m and the t coordinate of point K is 0.0050 s.

Part G

What is the period T?

Express your answer in seconds.

Part H

How much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement?

Express your answer in seconds.

Part I

What distance d does the object cover during one period of oscillation?

Express your answer in meters.

Part J

What distance d does the object cover between the moments labeled K and N on the graph?

Express your answer in meters.

The figure shows three graphs, A, B, C, which collectively describe the motion of an object in one dimension. The horizontal axis of each graph is time (symbol t). The vertical axes of the graphs are different. For one graph the vertical axis is position (symbol x), for another, it is the velocity (symbol v), and for another, it is acceleration (symbol a). Where the axes across is the point of zero for both time and the quantity corresponding to the vertical axis. Part A: For which sequence of graphs is the vertical coordinate position, velocity, and acceleration, respectively? (e.g., If you think graph Y is position versus time, graph X is velocity versus time, and graph Z is acceleration versus time, then your answer is Y, X, Z.). Multiple choice: 1)C, A, B 2)A, C, B 3)C, B, A 4)B, C, A 5)B, A, C 6)A, B, C. Part (b) Which one of the following situations can be most accurately represented by the graph, consistent with your answer to part (a)? (1) A car driving in the negative x-direction suddenly shifts into reverse and, with tires spinning, slows to a momentary stop before beginning to move in the position x-direction. (2)A car driving in the position x-direction slams on its brakes and comes to rest. (3) A rock is dropped from the rest from the top of a tall building. Assume air drag can be neglected, in which case its speed increases by the same amount each second. Take upward to be the position x-direction. (4) A sprinter runs a 100-meter dash and she runs in the negative x-direction. Starting from rest, she has a constant acceleration for the first 50 meters, then a constant speed for the remaining 50. Multiple choices: 1)4, 2)3, 3)2, 4)1.

Huygens' principle (first described by the Dutch scientist Christiaan Huygens in 1678) uses geometry to determine the shape of a wavefront at a time t, given some initial wavefront at an earlier time. This can be constructed by imagining that the initial wavefront is a source of wavelets that propagate from each point at the speed of light. From this principle, one can determine the angles of reflection and refraction by considering the speed of light at each point on the wavefront. In this problem, we will explore some of these concepts.

Part A

First, let's look at a plane wave that is incident on a flat piece of material with an index of refraction n. Part of the light is transmitted through the material and part of it is reflected. For the moment, let's just look at the part that is transmitted. At time t=0, the wavefront is a distance d away from the surface of the material. At time t=t₁, the wavefront is at the position of the material interface. Use the Huygens' principle to determine how far into the material (d') the wavefront has propagated by time t=2t₁.

Express your answer in terms of the variables n, t₁, and the speed of light in a vacuum c.

Note that there would be no change in the direction of the wavefront at any point because the wavefront encountered the material interface at the same time at all points. Thus, all of the individual wavefronts all propagated at the same speed, thereby maintaining the flat wavefront.

Now, instead of having a flat wavefront propagating normal to the material interface, we have a flat wavefront propagating toward the material at an angle of 55° to the axis perpendicular to the material interface. In this part, we will look at the relative positions of a few points—A, B, and C—on the wavefront to illustrate Huygens' principle. (Intro 1 figure) Point C touches the vacuum/material interface at time t=0 whereas point B is a distance d and point A is a distance 2d away from the interface.

Part B

What is the time t_B it will take for point B of the wavefront to encounter the vacuum/material interface?

Express your answer numerically to two decimal places in units of t₁ (the time it takes light to travel a distance d in a vacuum).

It should be noted that the accuracy of the method increases the smaller the distance is before making a new wavefront and redoing the wavelets. If you tried to just draw a large wavelet from point B, then it would hit the surface after only traveling a distance d. This large wavelet would make it difficult to visualize how to add up all of the other wavelets to make a coherent wavefront. As a result, one should just draw a line perpendicular to the wavefront at point B until it hits the surface, at which point the wavelets will change owing to the new index of refraction.

Part C

How far did point C go into the material interface in the time t=t_B that it took for point B to get to the interface? For this part we are looking for the distance traversed by the point C in the material (d_C).

Give your answer in terms of the time t_B, c, and n.

Part D

What is the new angle θ at which point C of the wavefront is propagating (relative to a line perpendicular to the vacuum/material interface)? Try to use the fact that you have a spherical wavefront propagating from t=0 at the point where C met the vacuum/material interface until time t_B when the wavefront at point B reached the interface.

Express your answer in terms of inverse trigonometric functions and n.