Q: Which of the following best explains why the simple pendulum exhibits SHM only for small angles?

For small angles, $$\sin \theta \approx \theta$$, making the restoring force proportional to displacement.

Question 1

Which of the following best describes the condition for simple harmonic motion (SHM) in a mass-spring system?

Accepted Answer

The restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.

Question 2

A block of mass $$m$$ attached to a spring with force constant $$k$$ oscillates with amplitude $$A. $$What is the expression for the total mechanical energy of the system?

Accepted Answer

$$E = \frac{1}{2} k A^2$

Question 3

The displacement of a particle in SHM is given by $$x = A \cos(\omega t). $$What is the velocity of the particle as a function of time?

Accepted Answer

$$v = -A \omega \sin(\omega t)$$

Question 4

For a mass-spring system undergoing SHM, which of the following correctly gives the angular frequency $$\omega$$?

Accepted Answer

$$\omega = \sqrt{\frac{k}{m}}$$

Question 5

A block attached to a spring oscillates with period $$T. If $$the mass is quadrupled while the spring constant remains the same, what happens to the period?

Accepted Answer

The period doubles.

Question 6

Which of the following equations correctly relates the period $$T$$ and angular frequency $$\omega$$ for SHM?

Accepted Answer

$$T = \frac{2\pi}{\omega}$$

Question 7

A simple pendulum of length $$l$$ swings with small amplitude. What is the period of oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{l}{g}}$$

Question 8

In SHM, when is the kinetic energy of the oscillating mass at its maximum value?

Accepted Answer

When the displacement is zero.

Question 9

A block oscillates on a frictionless surface with amplitude $$A$$ and period $$T. At $$what displacement $$x is $$the speed of the block half its maximum value?

Accepted Answer

$$x = \frac{A}{\sqrt{2}}$$

Question 10

Which of the following statements about the energy in SHM is correct?

Accepted Answer

The total energy is constant and is the sum of kinetic and potential energies.

Question 11

A simple pendulum has a length of $$1.0\ \mathrm{m}. $$What is its period near the Earth's surface ($$g = 9.8\ \mathrm{m/s^2$}$$)?

Accepted Answer

$$T = 2\pi \sqrt{\frac{1.0}{9.8}} \approx 2.01\ \mathrm{s}$$

Question 12

The acceleration of a particle in SHM is given by $$a = -\$$omega^2$$ x. $$What does the negative sign indicate?

Accepted Answer

The acceleration is directed opposite to the displacement from equilibrium.

Question 13

A mass $$m$$ attached to a spring oscillates with amplitude $$A. At $$what position is the potential energy equal to the kinetic energy?

Accepted Answer

$$x = \frac{A}{\sqrt{2}}$$

Question 14

Which of the following is a correct expression for the maximum speed of a mass in SHM?

Accepted Answer

$$v_{max} = A \omega$$

Question 15

A simple pendulum is released from a large angle. Which of the following is true about the period compared to the small-angle approximation?

Accepted Answer

The period is longer than predicted by the small-angle formula.

Question 16

Which of the following best explains why the simple pendulum exhibits SHM only for small angles?

Accepted Answer

For small angles, $$\sin 	heta \approx 	heta$$, making the restoring force proportional to displacement.

Question 17

A block attached to a spring is moving in SHM on a frictionless surface. If the amplitude is doubled, what happens to the total energy of the system?

Accepted Answer

The total energy is quadrupled.

Question 18

The projection of uniform circular motion onto a diameter of the circle is equivalent to:

Accepted Answer

Simple harmonic motion.

Question 19

A block of mass $$m is $$attached to a spring and oscillates with period $$T. If $$the spring constant $$k is $$increased by a factor of 4, what happens to the period?

Accepted Answer

The period is halved.

Question 20

For a simple pendulum, which of the following expressions gives the angular frequency $$\omega$$ for small oscillations?

Accepted Answer

$$\omega = \sqrt{\frac{g}{l}}$$

DR1- Simple Harmonic Motion

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