PHY 151_ Introduction to Physics

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```					• Remember: “Practice HW #11” posted on WebAssign
(0 points, covers material after HW #10)
Solutions will be posted on Friday afternoon

• Last Time: Hooke’s Law, Simple Harmonic Motion

• Today: SHM Position, Velocity, Acceleration;
Pendulum Motion

1
Review: SHM Period & Frequency
For SHM, relations between period, frequency, and angular
frequency for SHM :

time for one    number of oscillations   If think of one oscillation
oscillation     per second [Hz]         as corresponding to 2π
[seconds]                               radians, ω = number of

2
Conceptual
An object of mass m is attached to a horizontal spring, stretched
to a displacement of A from equilibrium, and then released.

It then undergoes harmonic oscillations on a frictionless surface
with period T0 . This is then repeated with a new object of mass
4m. What is its new period of oscillation?

(a) 2T0        (b) T0         (c) T0/2       (d) T0/4

3
Example
A 0.326-kg object is attached to a spring and executes SHM with a
period of 0.25 s. If the total energy of the system is 5.83 J, find:

(a) the maximum speed of the object,

(b) the force constant of the spring,

(c) the amplitude of the motion.

4
Position, Velocity, Acceleration vs. Time
Again, we will use the close mathematical relationship between
Circular Motion and SHM :
At some time, the x-position is:

ω
Suppose at t = 0, θ = 0. Then
we have θ = ωt, so :
θ

A                          But: ω = 2π/T = 2πf

5
Position, Velocity, Acceleration vs. Time
Displacement x :

Velocity v :

Velocity v :

6
Warning !!
Displacement x :

In these formulas, ωt is in
make sure your calculator is   Velocity v :
degrees when calculating
sines and cosines !!
Velocity v :

7
Demo: SHM Motion IS Sinusoidal !

8
Conceptual Question
If the amplitude A of a system undergoing SHM is doubled,
which of the following quantities does NOT change?

(a) Total energy

(b) Maximum speed

(c) Maximum acceleration

(d) Period

9
Conceptual Question
Suppose the position of an object moving with SHM is given by:
x = 4 cos (6πt), where x is in meters, and t is in seconds.

What is the period of this oscillating system?

(a) 4 s

(b) 1/6 s

(c) 1/3 s

(d) 6π s

(e) not enough information
10
Example: 13.28 (modified)
The position of an object connected to a spring varies with time
according to

x = A sin (Bt),   where: A = 0.052 m, B = 8π 1/s

(a) What is the period and frequency of this motion?

(b) What is the amplitude of this motion?

(c) Find the first time after t = 0 that the object reaches
x = 0.026 m.

(d) At this time, what is the object’s velocity and acceleration?

11
Motion of a Pendulum

When a pendulum swings back
and forth, is the motion SHM?

To answer this, we need to
examine the restoring force,
the force of gravity, that acts
along the circular arc.

Key Point: Mass on a spring
moves in 1-D only. Here, we
have motion in 2-D. But we
will consider “small
oscillations”.
12
Motion of a Pendulum

Let’s see if we can find a “Hooke’s
Law” for a pendulum …

End results …

13
So we found :

This implies :

• Period of a pendulum does not depend on its mass
• Period of a pendulum does not depend on its amplitude
(provided we are considering “small oscillations”)

• What is the length of the pendulum in our lecture hall ?
14
Conceptual Question
If a pendulum clock is tuned (i.e., its length is set) so that it
keeps perfect time at the base of a very tall mountain, will it also
keep perfect time when it is moved to the top of this mountain?

15
Example: 13.39

Earth/Mars Comparison

~15% of Earth’s volume
~11% of Earth’s mass

The free-fall acceleration on Mars is 3.7 m/s2.

(a) What length of pendulum has a period of 1 s on Earth? On Mars?

(b) An object is suspended from a spring with k = 10 N/m. What
mass would result in a period of 1 s on Earth? On Mars?
16
Next Class
• 13.7 – 13.8 : Intro to Properties of Waves (PHY 213),
Review for Final

17

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