PHYS 1300 Lecture Notes - Lecture 23: Angular Frequency, Simple Harmonic Motion, Standing Wave

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salmonbuffalo710

1 Feb 2017

School

University of Guelph

Department

Physics

Course

PHYS 1300

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Document Summary

Periodic/harmonic motion: any motion that repeats at regular intervals. Simple harmonic motion: the driving force depends on the displacement from equilibrium. Light waves are oscillations of electric and magnetic fields. Ripples on a pond are oscillations of water molecules. Pm - characterized by t; period, time per cycle. ; f - frequency, cycles per unit time f = 1 / t. Shm mathematically; x = -acos( t), where = 2 f (radians/s) A = 0. 200m, since 20. 0cm is the maximum/minimum. t = 10. 0s, completing 20 cycles. T = 0. 500s: a = 0. 200m, t = 0. 500s. It is a disturbance which is free to move through space carrying energy without the overall forward movement of. Travelling waves mathematically; y(x,t) = a(sin(2 (t/t x/( )) or = asin( t kx) Plus or minus is according to the direction of travel, + is the negative direction, - is the positive direction. A is the maximum displacement from the equilibrium.

Related Questions

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.

(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

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

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

(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.

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PHYS 1300 Lecture Notes - Lecture 23: Angular Frequency, Simple Harmonic Motion, Standing Wave

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