Questions: 6. Objects of various weights are hung from a rubber band that is suspended from a hook. The weights of the objects are plotted on a graph against the stretch of the rubber band. How can you tell from the graph whether the rubber band obeys Hooke's law? 7. How must the length of a pendulum be changed to double its period? How must the length be changed to halve the period? 8. If a car's wheel is out of balance, the car will shake strongly at a specific speed but not at a higher or lower speed. Explain. 9. How is uniform circular motion similar to simple harmonic motion? How are they different? Periodic Motion Lesson 1 1

6. Objects of various weights are hung from a rubber band that is suspended from a hook. The weights of the objects are plotted on a graph against the stretch of the rubber band. How can you tell from the graph whether the rubber band obeys Hooke's law?

Hooke's Law and Graph Shape

Hooke's law states that the force (weight in this case) is directly proportional to the extension (stretch). Mathematically, this is \(F = kx\), where \(F\) is the force, \(x\) is the extension, and \(k\) is the spring constant.

Determining if Hooke's Law is Obeyed

A graph of force against extension that obeys Hooke's law will be a straight line passing through the origin. The slope of this line is equal to the spring constant, \(k\). If the graph is not a straight line, or if it doesn't pass through the origin, then the rubber band is not obeying Hooke's law.

\(\boxed{\text{The graph will be a straight line passing through the origin if the rubber band obeys Hooke's law.}}\)

7. How must the length of a pendulum be changed to double its period? How must the length be changed to halve the period?

Relationship between Length and Period

The period \(T\) of a simple pendulum is given by the formula \(T = 2\pi\sqrt{\frac{L}{g}}\), where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity. This shows that the period is proportional to the square root of the length.

Doubling the Period

To double the period, we need to increase \(L\) such that \(2T = 2\pi\sqrt{\frac{L'}{g}}\), where \(L'\) is the new length. Squaring both sides and comparing to the original equation, we find \(L' = 4L\). Therefore, the length must be quadrupled.

Halving the Period

To halve the period, we want \(\frac{T}{2} = 2\pi\sqrt{\frac{L''}{g}}\). Squaring both sides and comparing gives \(L'' = \frac{L}{4}\). So we need to reduce the length to one-fourth of its original value.

\(\boxed{\text{To double the period, quadruple the length. To halve the period, quarter the length.}}\)

8. If a car's wheel is out of balance, the car will shake strongly at a specific speed but not at a higher or lower speed. Explain.

Resonance and Natural Frequency

Every object has a natural frequency at which it vibrates most readily. When a force is applied to the object at its natural frequency, resonance occurs, causing large amplitude vibrations.

Unbalanced Wheel and Resonance

An unbalanced wheel creates a periodic force as it rotates. At a certain speed, the frequency of this force matches the natural frequency of the car's suspension system. This leads to resonance and the car shakes violently. At higher or lower speeds, the forcing frequency is different, and resonance doesn't occur, resulting in less shaking.

\(\boxed{\text{At a specific speed, the rotation frequency of the unbalanced wheel matches the car's natural frequency, causing resonance and strong vibrations.}}\)

\(\boxed{\text{1. The graph of force vs. extension will be a straight line through the origin if Hooke's law is obeyed.}}\) \(\boxed{\text{2. To double the period, quadruple the length. To halve the period, reduce the length to one-fourth.}}\) \(\boxed{\text{3. Resonance between the unbalanced wheel's rotation frequency and the car's natural frequency causes shaking at a specific speed.}}\)