Q1. What is the differential equation of motion for a mass \( m \) attached to an ideal spring with spring constant \( k \)?

\( m\ddot{x} + kx = 0 \)
\( m\dot{x} + kx = 0 \)
\( k\ddot{x} + mx = 0 \)
\( \ddot{x} + mkx = 0 \)

Q2. The general solution to \( \ddot{x} + \omega^2 x = 0 \) is:

\( x(t) = A \cos(\omega t + \phi) \)
\( x(t) = A \exp(-\omega t) \)
\( x(t) = A \cos(\omega t) + B \sin(\omega t) \)
Both A and C

Q3. For a mass-spring system, the natural angular frequency is:

\( \omega = \sqrt{\frac{k}{m}} \)
\( \omega = \frac{k}{m} \)
\( \omega = \frac{m}{k} \)
\( \omega = \sqrt{\frac{m}{k}} \)

Q4. Which quantity remains conserved in the ideal mass-spring system?

Kinetic energy
Potential energy
Total mechanical energy
Linear momentum

Q5. If the mass is displaced by \( x_0 \) and released, what is the amplitude of oscillation?

\( 0 \)
\( x_0 \)
\( 2x_0 \)
\( \sqrt{x_0} \)

Q6. The energy stored in a spring at maximum extension \( A \) is:

\( \frac{1}{2} m A^2 \)
\( \frac{1}{2} k A^2 \)
\( k A \)
\( mA \)

Q7. For a simple pendulum of length \( l \), the linearized equation of motion is:

\( \ddot{\theta} + \frac{g}{l} \sin(\theta) = 0 \)
\( \ddot{\theta} + \frac{g}{l} \theta = 0 \)
\( \ddot{x} + \frac{k}{m} x = 0 \)
\( \dot{\theta} = \theta \)

Q8. For small angles, the simple pendulum exhibits:

Damped oscillations
Chaotic motion
Simple harmonic motion
Constant angular velocity

Q9. The frequency of oscillation of a simple pendulum (small angle) is:

\( \frac{1}{2\pi} \sqrt{\frac{k}{m}} \)
\( \frac{1}{2\pi} \sqrt{\frac{g}{l}} \)
\( \sqrt{g l} \)
\( \frac{1}{2\pi} \sqrt{gl} \)

Q10. In SHM, the velocity is maximum when:

Displacement is maximum
Displacement is zero
Acceleration is zero
Both B and C

Q11. What is the total mechanical energy of a mass-spring system in SHM with amplitude \( A \)?

\( \frac{1}{2} k x^2 \)
\( \frac{1}{2} m v^2 \)
\( \frac{1}{2} k A^2 \)
\( \frac{1}{2} k A \)

Q12. A block of mass \( m \) is attached to a spring and oscillates with angular frequency \( \omega \). What is the period?

\( 2\pi \omega \)
\( 2\pi \sqrt{m/k} \)
\( \omega/2\pi \)
\( 2\pi \sqrt{k/m} \)

Q13. For a pendulum released from rest at angle \( \theta_0 \), the restoring torque is:

\( mg\cos(\theta) \)
\( -mg\ell\sin(\theta) \)
\( mg\ell\cos(\theta) \)
\( -mg\theta \)

Q14. If \( x(t) = A \cos(\omega t + \phi) \), then the maximum velocity is:

\( A\omega \)
\( A/\omega \)
\( \omega^2 A \)
\( \omega/A \)

Q15. A damped spring system is governed by:

\( m\ddot{x} + c\dot{x} + kx = 0 \)
\( m\ddot{x} + kx = 0 \)
\( m\ddot{x} + k\dot{x} + cx = 0 \)
\( m\ddot{x} + cx = 0 \)

Q16. In damped motion, what does the damping coefficient \( c \) represent?

Restoring force
Energy dissipation rate
Spring constant
Mass of system

Q17. What type of solution is expected when damping is small?

Exponentially growing
Critically damped
Underdamped oscillation
No motion

Q18. In forced oscillation, the steady-state amplitude is maximum when:

Driving frequency is zero
Driving frequency matches natural frequency
Damping is large
Spring is relaxed

Q19. Which term in the damped oscillator equation causes energy loss?

\( m\ddot{x} \)
\( c\dot{x} \)
\( kx \)
\( \ddot{x} \)

Q20. The solution to \( \ddot{x} + 2\beta \dot{x} + \omega^2 x = 0 \) is oscillatory when:

\( \beta > \omega \)
\( \beta = \omega \)
\( \beta < \omega \)
All of the above