Physics

Oscillations and Simple Harmonic Motion

138 Questions

Oscillations and simple harmonic motion focus on amplitude, damped vibrations, and force equations. These physics principles are essential for various engineering and civil services examinations. Review these problems to understand the core mechanics of oscillating bodies.

Damped harmonic oscillationSHM amplitudeForce equationsForced oscillationVibratory motion concepts

Oscillations and Simple Harmonic Motion Questions

Multiple choice physics wave motion wave velocity speed and acceleration of travelling wave speed of a travelling wave

A particle moves with simple harmonic motion in a straight line. In first $\tau s,$, after starting from rest it travels a distance $a$, and in next $\tau s$ it travels $2a$, in same direction, then:

  1. amplitude of motion is $4a$
  2. time period of oscillations is $6$,
  3. amplitude of motion is $3a$$\tau $
  4. time period of oscillations is $8$,$\tau $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For SHM starting from rest at x = A, x(t) = A cos(omega * t). Distance traveled in time tau is A - A cos(omega * tau) = a. In next tau, distance is A cos(omega * tau) - A cos(2 * omega * tau) = 2a. Solving these equations leads to the amplitude being 4a.

Multiple choice physics wave motion wave velocity speed and acceleration of travelling wave speed of a travelling wave

The displacement from the position of equilibrium of a point $4\ cm$ from a source of sinusoidal oscillations is half the amplitude at the moment $t=\dfrac{T}{6} (T$ is the time period$)$. Assume that the source was at mean position at $t=0$. The wavelength of the running wave is 

  1. $0.96\ m$
  2. $0.48\ m$
  3. $0.24\ m$
  4. $0.12\ m$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Going by the data given to us, this wave is sinusoidal in nature and the wave equation takes the form of
$y = A\sin( \omega t - kx),$ as it is given that at $t = 0,$  the source is at mean position.
Here$,\ x = 4\ cm = 0.04\ m$
$y = A/2$
Amplitude $= A$
$t = \dfrac{T}{6}$
We know that $ \omega  = 2 \dfrac{ \pi }{T}$
$\Rightarrow \dfrac{A}{2} = A\sin((2  \pi  / T)(T/6) - 0.04k)$
$\sin((2 \pi  / T)(T/6) - 0.04k) = 1/2$
$\Rightarrow ((2  \pi / T)(T/6) - 0.04k) =  \pi / 6$
$k =  \pi  / 0.24$
wavelength $ \lambda = 2\pi  / k = 0.48\ m$

Multiple choice physics wave motion wave velocity speed and acceleration of travelling wave speed of a travelling wave

The vibration of a string of length 60 cm fixed at both ends are represented by $ y=4sin (\frac { \pi x}{15}) cos (96 \pi t) $ where x and y are in cm and t in second. the particle velocity at x=7.5 cm and t=0.25 s is

  1. Zero
  2. $ 10 cm s^{-1} $
  3. $ 100 cm s^{-1} $
  4. $ (4 \times 96) cm s^{-1} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The particle velocity is given by the partial derivative of y with respect to t. Since the wave is a standing wave, the velocity is v = dy/dt = -4 * 96 * pi * sin(pi*x/15) * sin(96 * pi * t). At t = 0.25 s, sin(96 * pi * 0.25) = sin(24 * pi) = 0, so the velocity is zero.

Multiple choice physics wave motion wave velocity speed and acceleration of travelling wave speed of a travelling wave

A standing wave of time period T is set up in string clamped between two rigid supports at t=0 antitode is at its maximum displacement A

  1. The energy of a node is equal to energy of an anitode for the first time at t=T/8

  2. The energy of node and antitode becomes equal after every T/2 second.
  3. the displacement of the particle of antinode at $ t= \frac {T}{8} is \sqrt 2 A $

  4. The displacement of the particle of node is zero
Reveal answer Fill a bubble to check yourself
C Correct answer
Multiple choice

A simple harmonic oscillator is a system that oscillates about an equilibrium position with a constant amplitude. The equation for the displacement of a simple harmonic oscillator is:

  1. x = Acos(ωt + φ)

  2. x = Fd

  3. x = kx
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation for the displacement of a simple harmonic oscillator is x = Acos(ωt + φ), where x is the displacement, A is the amplitude of the oscillation, ω is the angular frequency, t is the time, and φ is the phase angle.

Multiple choice

A damped harmonic oscillator is a system that oscillates about an equilibrium position with a decreasing amplitude. The equation for the displacement of a damped harmonic oscillator is:

  1. x = Ae^(-γt)cos(ωt + φ)

  2. x = Fd

  3. x = kx
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation for the displacement of a damped harmonic oscillator is x = Ae^(-γt)cos(ωt + φ), where x is the displacement, A is the amplitude of the oscillation, γ is the damping coefficient, ω is the angular frequency, t is the time, and φ is the phase angle.

Multiple choice

What is the Hamilton-Jacobi equation in classical mechanics?

  1. A partial differential equation that is used to solve the equations of motion for a system.

  2. A partial differential equation that is used to solve the Poisson equation.

  3. A partial differential equation that is used to solve the Laplace equation.

  4. A partial differential equation that is used to solve the Helmholtz equation.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Hamilton-Jacobi equation is a partial differential equation that is used to solve the equations of motion for a system, and it is equivalent to the Euler-Lagrange equation.

Multiple choice

In a simple harmonic motion, the total mechanical energy of the system is:

  1. Constant

  2. Increasing

  3. Decreasing

  4. Zero
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a simple harmonic motion, the total mechanical energy (kinetic + potential) remains constant throughout the motion.

Multiple choice

What happens when the frequency of the driving force is equal to the natural frequency of the system?

  1. The system oscillates with greater amplitude.

  2. The system oscillates with smaller amplitude.

  3. The system oscillates with the same amplitude.

  4. The system does not oscillate at all.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

When the frequency of the driving force is equal to the natural frequency of the system, the system oscillates with greater amplitude. This is because the driving force is able to transfer more energy to the system than it can at other frequencies.

Multiple choice

In forced harmonic motion, the amplitude of the oscillation is maximum when the driving frequency is equal to the:

  1. Natural frequency

  2. Resonant frequency

  3. Damping frequency

  4. Angular frequency
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

At the resonant frequency, the system's natural frequency matches the driving frequency, leading to maximum amplitude.

Multiple choice

Which of the following is NOT a characteristic of forced harmonic motion?

  1. Amplitude

  2. Frequency

  3. Phase shift

  4. Damping
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Damping is not a characteristic of forced harmonic motion but rather a factor that affects the system's behavior.

Multiple choice

The equation of motion for a forced harmonic oscillator is given by: (m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0\cos(\omega t)). What does (F_0) represent?

  1. Mass of the oscillator

  2. Damping coefficient

  3. Spring constant

  4. Driving force amplitude
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

(F_0) represents the amplitude of the driving force applied to the oscillator.

Multiple choice

What is the relationship between the phase angle (\phi) and the time lag (\Delta t) in forced harmonic motion?

  1. \(\phi = \Delta t\)
  2. \(\phi = \frac{\Delta t}{2\pi}\)
  3. \(\phi = 2\pi\Delta t\)
  4. \(\phi = \frac{2\pi}{\Delta t}\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The phase angle (\phi) is related to the time lag (\Delta t) by the equation (\phi = \frac{\Delta t}{2\pi}).

Multiple choice

What is the effect of increasing the driving force frequency on the phase angle (\phi) in forced harmonic motion?

  1. \(\phi\) increases
  2. \(\phi\) decreases
  3. \(\phi\) remains constant
  4. \(\phi\) becomes zero
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

As the driving force frequency increases, the phase angle (\phi) also increases.

Multiple choice

Which of the following is NOT a characteristic of the steady-state response in forced harmonic motion?

  1. Constant amplitude

  2. Constant frequency

  3. Constant phase angle

  4. Constant energy
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Energy is not constant in the steady-state response of forced harmonic motion due to energy dissipation.