Simple Harmonic Motion (SHM)
- repetitive movement back and forth through an equilibrium, or central, position,
- max. displacement on one side of this position is = max. displacement on the other side.
- Force on the object is proportional to the distance to the equilibrium position
I. Vibrating Mass on a Spring
$$KE_1 + PE_1 = KE_2 + PE_2$$
Where is the acceleration the greatest? Why?
Smallest? Why?
Teacher CER Answer Key
Reasoning: Acceleration is proportional to displacement. Maximum at amplitude, zero at equilibrium.
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II. Period
Period - time for one full
determined by .... and
$$T = 2\pi\sqrt{\frac{m}{k}}$$
III. Energy Changes – Conservation of Energy
PE at C equals .....
Ex) Mass M is attached to a spring with a spring constant k. If the maximum displacement of a mass M from its equilibrium position is A, find the velocity of the mass at B:
Position as a function of time
$X = A \cos \theta$, [ when $t = 0, X = A$ ]
Since we can replace $\theta$ with $\omega t$.
$$X = A \cos(\omega t)$$
Where A is amplitude, T is period, and t is time.
$X = A \cos([\text{______}]t)$
$X = A \cos([\text{_____}]t)$
Graph Analysis
What is the Amplitude? Period?
Sample Problems
Do Problems 1 - 5, 8 - 12
1. Where is the block located when its velocity is a maximum in magnitude?
A) x = 0
B) x = ±A
C) x = +A/2
D) x = -A/2
2. Where is the block located when its potential energy is a maximum?
A) x = 0
B) x = ±A
C) x = +A/2
D) x = -A/2
3. Where is the block located when its acceleration is a minimum in magnitude?
A) x = 0
B) x = ±A
C) x = +A/2
D) x = -A/2
4. What is the period of the system if the amplitude is doubled?
5. What is the period of the system when it is located on the Moon?
A) 6T
B) T/6
E) T
D) $T/\sqrt{6}$
Free Response Problems
1. Potential Energy Graph Analysis
A 0.4 kg object is attached to a horizontal spring undergoes SHM with the total energy of 0.2 J. The potential energy as a function of position presented by the graph below:
- What is the amplitude of oscillations?
- What is the spring constant?
- What is the kinetic energy of the system at point x = 2.5 cm?
- Indicate point or points where the kinetic energy equals the potential energy of the system.
- What is the maximum speed of the object?
2. Kinetic Energy Graph Analysis
A 0.20 kg object attached to a horizontal spring undergoes SHM with total energy of 0.40 J. The kinetic energy as a function of position presented by the graph below:
- What is the maximum displacement from equilibrium?
- What is the maximum speed of the object?
- What is the spring constant?
- Indicate point or points where the kinetic energy equals the potential energy of the system.
- What is the potential energy of the system at point x = 2 cm?
S.H.M. – Pendulum
(T) Period of a Pendulum
- Determined by ....
- Equation
- Frequency
Finding the height of a pendulum with a length of L and an angle $\theta$:
$$T_p = 2\pi\sqrt{\frac{L}{g}}$$
Pendulum MCQ Set
1. If the mass of the pendulum is doubled what is the new period of the pendulum?
A) T/2
B) 2T
C) T
D) $\sqrt{2}T$
2. If the length of the pendulum is doubled what is the new period of the pendulum?
A) T/2
B) 2T
C) T
D) $\sqrt{2}T$
3. What is the length of a simple pendulum if it oscillates with a period of 2.0 s?
A) 2.0 m
B) 1.0 m
C) 0.5 m
D) 0.1 m
4. Displaced up by a distance 0.2 m. What is the maximum speed?
A) 1 m/s
B) 2 m/s
C) 3 m/s
5. Speed of the ball at the lowest point?
A) $\sqrt{2gL}$
B) $\sqrt{2gL(1-\cos\theta)}$
C) $\sqrt{gL}$
Free Response: Collisions
3. Clay-Pendulum Collision
A 20. g piece of clay moving at a speed of 50. m/s strikes a 500. g pendulum bob at rest. The length of a string is 0.80 m. After the collision, the clay-bob system starts to oscillate as a simple pendulum.
- What is the speed of the clay-bob system after the collision?
- What is the maximum angular displacement of the pendulum?
- What is the period of the clay-bob oscillating system?
- What is the total energy of the oscillating system?
The pendulum bob makes one complete oscillation and the string breaks at the lowest point.
- What is the maximum horizontal distance of the bob when it strikes the floor 0.70 m below?
4. Inelastic Spring Collision
A small block moving with a constant speed v collides inelastically with a block M attached to a spring k. Find speed and amplitude after collision:
Instructional Video II & Wave Analysis
6. A second identical spring k is added in parallel. What is the period of oscillations?
A) 2T
B) $T/\sqrt{2}$
C) $\sqrt{2}T$
7. If the mass in each system is doubled, which of the following is true about the new period?
| Mass-spring | Simple pendulum |
|---|
C) $\sqrt{2}T$ |
$T$ |
A) $T$ |
$T$ |
8. Formula: $x = (0.1\text{ m})\sin(4\pi t)$. What is the period?
9. Formula: $x = (0.5\text{ m})\cos(\pi t)$. What is the amplitude?
10. Identify velocity and acceleration at the time 1.5 s from graph below:
D) $v > 0, a = 0$
B) $v = 0, a = 0$
11. Which is true about the amplitude and period from graph below?
| Amplitude | Period |
|---|
D) 1 m |
0.8 s |
A) 2 m |
0.4 s |
12. An object oscillates at the end of a spring. The position as a function of time is presented by the graph. Which of the following formulas represent the position and velocity of the object?
A) $x = (0.5)\sin(\pi t)$
$v = (0.5\pi)\sin(\pi t)$
B) $x = (0.5)\sin(\pi t)$
$v = (0.5\pi)\cos(\pi t)$
C) $x = (0.5)\cos(\pi t)$
$v = (0.5\pi)\sin(\pi t)$
D) $x = (0.5\pi)\sin(\pi t)$
$v = (0.5)\sin(\pi t)$
Teacher CER Answer Key
Final Answers: 10:D, 11:D, 12:B.
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