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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 6.12 Practice, Study, and Apply - Oscillations

Subsection Practice

Calculation 6.12.1. Two Springs and a Mass.

A mass is compressed between two springs on a frictionless horizontal surface. When the mass is in equilibrium, both springs are at their relaxed length. Which of the following statements are true regarding this situation?
  1. The mass is traveling its fastest when the force by the springs is the greatest.
  2. The mass accelerates the most when it’s furthest from its equilibrium position.
  3. The period of the mass changes depending on the value of its maximum displacement.
  4. You could replace those two springs with one spring whose constant is equal to \(k_1 + k_2\) and the frequency of the oscillation would not change.
Answer.
B., D.

Calculation 6.12.2. Pendulum on Jupiter.

A mass on an ideal spring and a simple pendulum have the same oscillation frequency on Earth. On Jupiter’s moon Europa, the gravitational acceleration is about 1/9 that of Earth’s. How would the frequency of the pendulum compare to the frequency of the mass-spring system if they were both oscillating on Europa?
  1. The pendulum frequency would be 1/9 the mass-spring frequency.
  2. The pendulum frequency would be 1/3 the mass-spring frequency.
  3. The pendulum frequency would still be the same as the mass-spring frequency.
  4. The pendulum frequency would be 3 times the mass-spring frequency.
  5. The pendulum frequency would be 9 times the mass-spring frequency.
  6. None of the above are true.
Answer.
B.

Calculation 6.12.3. Bird on a Tree Branch.

On a windless day, Callie the Cardinal-Grosbeak (mass = 43 g) is swaying on a tree twig of negligible mass. Being quite the scientist, she notices that the tree branch behaves just like a spring; she records her position as being described by this time function: \(x(t) = (5 cm) cos(\frac{4\pi}{3}t)\text{.}\) What is the maximum acceleration Callie experiences?
  1. 5 \(m/s^2\)
  2. 0.05 \(m/s^2\)
  3. 0.209 \(m/s^2\)
  4. 0.877 \(m/s^2\)
  5. 3.67 \(m/s^2\)
What is the spring constant of the branch?
  1. 754 \(N/m\)
  2. 0.754 \(N/m\)
  3. 0.180 \(N/m\)
  4. 0.038 \(N/m\)
  5. 3.16 \(N/m\)
Answer 1.
D.
Answer 2.
B.

Calculation 6.12.4. Oscillating Mass.

A mass is connected to a spring and is set to oscillating. You start your stopwatch just as the mass passes the equilibrium position, traveling in negative x-direction. Which time function should be used to model the position of the mass?
  1. Positive Sine
  2. Negative Sine
  3. Positive Cosine
  4. Negative Cosine
Which time function should be used to model the velocity of the mass?
  1. Positive Sine
  2. Negative Sine
  3. Positive Cosine
  4. Negative Cosine
Which function should be used to model the acceleration of the mass as a function of time?
  1. Positive Sine
  2. Negative Sine
  3. Positive Cosine
  4. Negative Cosine
Which function should be used to model the net force acting on the mass as a function of time?
  1. Positive Sine
  2. Negative Sine
  3. Positive Cosine
  4. Negative Cosine
Answer 1.
B.
Answer 2.
D.
Answer 3.
A.
Answer 4.
A.

Subsection Study

A*R*C*S 6.12.5. Ant on a String.

A string (length \(L\)) with constant tension \(T\) is tied horizontally between two walls (this is effectively how strings in musical instruments work!). An ant (mass \(m\)) clings to the very center of the string, which is moving back and forth horizontally with amplitude \(A\) much smaller than \(L\text{.}\)
Model the situation as a simple harmonic oscillator, find the angular frequency, and create one or more graphs showing the motion of the system as a function of time.
Tip.
You may find it necessary to make an approximation in order to model the system as a simple harmonic oscillator!

Explanation 6.12.6. Weighted Pendulum.

You can make an interesting pendulum by taking a very light, stiff, cylindrical object (like a paper towel roll) of length L and taping or gluing two small, heavy objects to the inside: mass \(m_1\) at an end and mass \(m_2\) exactly in the center. The end with nothing attached to it is then attached to the ceiling with a very short piece of string, and the pendulum is allowed to swing back and forth with amplitude A much smaller than \(L\text{.}\)
In case A, suppose that \(m_1\) > \(m_2\text{,}\) and in case B, suppose that \(m_2\) > \(m_1\text{.}\) Will the period in case A be greater than, less than, or equal to the period in case B?

Subsection Apply

Explanation 6.12.7. Pendulum on a Tree.

You find a very tall tree one day and decide to measure the height of the tree using a pendulum. You tie a small rock to a rope that is also tied to a branch at the top of the tree, so that when the rope is vertical the rock is very close to the ground. You then release the rock from rest at an initial angle of 5°. You measure the amount of time it takes for the rock to return to your hand. Your friend performs the same experiment with a different tree, but uses a smaller initial angle of only 2.5°. Your friend measures the amount of time it takes for the rock to return to her hand to be half as big as the time you found. Is the height of your tree greater than, less than, or equal to the height of your friend’s tree?

A*R*C*S 6.12.8. Coin on a Block.

A block is attached to the top of a spring that is oscillating vertically. A small coin with negligible mass is riding on top of the block (it is not attached to either the block or the spring). You measure that the largest possible time for the block to complete one full oscillation, without causing the coin to fall off, is 2.5 s. What is the amplitude of this oscillation?
Tip.
For your representation (part 1c), sketch graphs of the position, velocity, and acceleration of the block vs. time.

References References

[1]
Practice activities provided by BoxSand: https://boxsand.physics.oregonstate.edu/welcome.
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