PHYS 1C Lecture Notes - Lecture 2: Simple Harmonic Motion, Propagation Constant, Net Force

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5 Apr 2018

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Let"s say we have a mass hanging at rest from a spring. The spring pulls up on the mass so that it does not accelerate. This spring force from the spring (also known as a restoring force, fspring) will resist wither a compression or a stretching. In general, each spring will have a different resistance to a certain displacement. Hooke"s law gives the value of the restoring force as a function of displacement where k is a constant of proportionality also known as the spring constant (units of k are [n/m]). The minus sign in hooke"s law is to show that the restoring force is opposite in direction to the displacement vector. Please not that the restoring force in hooke"s law is not constant, as we increase displacement the restoring force will increase linearly. Periodic motion is the motion of an object that regularly repeats. The object returns to a given position after a fixed time interval.

Related Questions

Use Hooke's Law for this (F = - k s ): Where F is the spring'srestoring force; k is the spring constant; and s is the stretch.The negative sign means the spring's restoring force is oppositethe stretch direction.
You have a plot from weight [N] versus stretch [m]. The data formsa linear trend y = 3.743 * x + 1.28. How much will the springstretch if 56.7 grams is hung on the spring?
[cm]
Answer in centimeters with three significant figures or N/A if notenough information is given to answer. When you calculate youransswer, don't use the negative sign in the Hooke's Law formula.Just know that the negative sign simply denotes the force directionis opposite the stretch (or compression).

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 1C Lecture Notes - Lecture 2: Simple Harmonic Motion, Propagation Constant, Net Force

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