PHYS 211 Study Guide - Midterm Guide: Wave Equation, Statics, Simple Harmonic Motion

 A mass, m, is attached to a spring of force constant 85.0 N/m and allowed to oscillate. The figure shows a graph of its velocity as a function of time, t.

·Find the period, frequency, angular frequency and amplitude for this motion.

·At what times between t=0 and t=1.8s does the mass reach maximum displacement from the equilibrium position?

·Find the magnitude of the maximum acceleration of the mass.

·At what times between t=0 and t=1.8s does the mass reach maximum acceleration (magnitude)?

·What is the mass, m?

·What is the maximum speed of the mass, and at which times between t=0 and t=1.8s does it attain that speed? 

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.

P2) A uniformly charged sphere of radius R has an electricfield
3
E r
R
= for r < R.
Consider a very simplified model of the atom where the positivenucleus, consisting of Z
protons, is a point charge at the center of the atom, surrounded bya uniform cloud of Z
electrons.
(a) If the nucleus is displaced relative to the electron cloud by asmall distance x, what is
the force acting on the nucleus due to the cloud (and equivalently,the force acting on the
cloud due to the nucleus)?
(b) In PHYS 132 you will have seen simple harmonic oscillators,such as a mass on a
spring. It can be shown that if the acceleration a of an object isdirectly proportional to
its displacement x from a point of equilibrium, then the objectwill undergo simple
harmonic motion with a frequency:
2
f ?
p
= , where 2
a x
?
=-
Determine the frequency of oscillation if the nucleus is displacedrelative to the center of
the electron cloud and then let go. Hint: from knowing the massesof the protons and
electrons, is it the cloud that primarily moves back and forth oris it the nucleus?
Calculate this frequency for a hydrogen atom; you can find theatomic radius R on
Wikipedia. If this oscillating atom emits an electromagnetic waveat the same frequency,
what is its wavelength?