Physics

Oscillations and Periodic Motion

162 Questions

Oscillations and periodic motion describe the movement of objects repeating their paths in regular intervals. Key concepts include simple pendulums, kinetic energy variations, and mechanical resonance. This physics topic is vital for various competitive exams.

Simple pendulumTime period calculationsKinetic energy in SHMMechanical resonanceDamped oscillations

Oscillations and Periodic Motion Questions

Multiple choice physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

The period of oscillation of a simple pendulum is Given by $ T=2\pi \sqrt { \frac { \ell  }{ g }  }$  where  $\ell$ is about 100 cm and is known to have 1 mm accuracy. The period is about 2 s. The time of 100 oscillation is measured by a stop watch of least count 0.1 s. The percentage error in g is:-

  1. 0.1 %

  2. 1 %

  3. 0.2 %

  4. 0.8 %
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

T = 2*pi*sqrt(L/g) implies g = 4*pi^2*L/T^2. Relative error dg/g = dL/L + 2*dT/T. dL = 0.1 cm, L = 100 cm, dL/L = 0.001. dT = 0.1/100 = 0.001 s, T = 2 s, dT/T = 0.0005. dg/g = 0.001 + 2(0.0005) = 0.002 or 0.2%.

Multiple choice physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

The time period of oscillation of simple pendulum given by $T = 2\pi\sqrt{\frac{L}{g}}$ where $L= (200 \pm 0.1) cm$. The time period, T =4s and the time of 100 oscillations is measured using a stopwatch of least count 0.1 s. The percentage error in g is

  1. $0.1$%
  2. $1.5$%
  3. $2$%
  4. $4$%
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

g = 4*pi^2*L/T^2. dg/g = dL/L + 2*dT/T. dL = 0.1, L = 200, dL/L = 0.0005. dT = 0.1/100 = 0.001, T = 4, dT/T = 0.00025. dg/g = 0.0005 + 2(0.00025) = 0.001 or 0.1%.

Multiple choice physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

While measuring the acceleration due to gravity by a simple pendulum, a student makes a positive error of 1% in the length of the pendulum and a negative error of 3% in the value of time period. His per-centre error in the n=measurement of g by the relation $g={ 4\pi  }^{ 2 }\left( I/T^{ 2 } \right) $ will be 

  1. 2%

  2. 4%

  3. 7%

  4. 10%
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given g = 4 * pi^2 * (L / T^2). The relative error is dg/g = dL/L + 2 * dT/T. Using magnitudes: 1% + 2 * 3% = 1% + 6% = 7%.

Multiple choice physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

The length of a pendulum is measured as $1.01$ m and time for $30$ oscillations is measured as $1$ minute $3$ seconds. Error in length is $0.01$ m and error in time is $3$ seconds. The percentage error in the measurement of acceleration due to gravity is:

  1. $1$
  2. $5$
  3. $10$
  4. $15$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

$\implies g=\dfrac{4\pi^2 l}{T^2}$

Thus $\dfrac{\Delta g}{g}=\dfrac{\Delta l}{l}+2\dfrac{\Delta T}{T}$

Percentage error in measurement of: 
$g=\dfrac{\Delta g}{g}\times 100=(\dfrac{\Delta l}{l}+2\dfrac{\Delta T}{T})\times 100$
   $=(\dfrac{0.01}{1.01}+2\dfrac{3}{63})\times 100$% $=10$%

Multiple choice physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

A students performs an experiment to determine the acceleration due to gravity (g) at a place using a simple pendulum. The length of the pendulum is 60 cm and the total time for 30 oscillations is 100s. What is maximum percentage error for the measurement g ? Given, least count for time $=0.1 s$ and least count for length $=0.1 cm$. 

  1. $0.26 \%$
  2. $0.3 \%$
  3. $0.36 \%$
  4. $3.6 \%$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a pendulum, the time period is: $T=2\pi\sqrt{\dfrac{l}{g}}$ 
$\Rightarrow g=\dfrac{4\pi^2 l}{T^2}=\dfrac{4\pi^2 ln^2}{t^2}$

Take $ln$ and then differentiate:
$\dfrac{\Delta g}{g}=\dfrac{\Delta l}{l}+2\dfrac{\Delta t}{t}$   (as n is constant so its derivative will be zero)

The % error in g $=\dfrac{\Delta g}{g}\times 100=\dfrac{\Delta l}{l}\times 100+2\dfrac{\Delta t}{t}\times 100=\dfrac{0.1}{60}\times 100+2\dfrac{0.1}{100}\times 100=0.36 \%$

Multiple choice physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

The percentage errors in the measurement of length and time period of a simple pendulum are 1% and 2% respectively. Then, the maximum error in the measurement of acceleration due to gravity is:

  1. $3%$
  2. $4%$
  3. $6%$
  4. $5%$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

g = 4 * pi^2 * L / T^2. Relative error dg/g = dL/L + 2 * dT/T. Given dL/L = 1% and dT/T = 2%, dg/g = 1% + 2 * 2% = 5%.

Multiple choice rotational equilibrium option b: engineering physics motion of system of particles and rigid bodies equilibrium physics

The bob of simple pendulum having length l, is displaced from mean position to an angular position with respect to vertical. If it is released, then velocity of bob at equilibrium position : 

  1. $\sqrt { 2g\ell (1-cos\theta ) }$
  2. $\sqrt { 2g\ell (1+cos\theta ) } $
  3. $\sqrt { 2g\ell cos\theta } $
  4. $\sqrt { 2g\ell } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Potential energy at extreme position = kinetic energy at mean position

$mg\ell (1-cos\theta )=\frac { 1 }{ 2 } m{ v }^{ 2 }$

Multiple choice

Which of the following is a type of nonlinear differential equation that is often used to model the motion of a pendulum?

  1. Simple pendulum equation

  2. Damped pendulum equation

  3. Driven pendulum equation

  4. All of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The simple pendulum equation, damped pendulum equation, and driven pendulum equation are all types of nonlinear differential equations that are often used to model the motion of a pendulum.

Multiple choice

A pendulum is a weight suspended from a pivot so that it can swing freely. The period of a pendulum is the time it takes for the pendulum to make one complete swing. The equation for the period of a pendulum is:

  1. T = 2π√(L/g)

  2. T = Fd

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

The period of a pendulum is the time it takes for the pendulum to make one complete swing. The equation for the period of a pendulum is T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

Multiple choice

Which differential equation is used to model the motion of a pendulum?

  1. Hooke's Law

  2. Newton's Second Law

  3. Simple Harmonic Motion Equation

  4. Damped Harmonic Motion Equation
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Damped Harmonic Motion Equation is used to model the motion of a pendulum, taking into account damping forces.

Multiple choice

A pendulum of length (L) is released from an angle (\theta_0). What is the equation of motion for the pendulum?

  1. \(m\frac{d^2\theta}{dt^2} + mg\sin\theta = 0\)
  2. \(m\frac{d^2\theta}{dt^2} - mg\sin\theta = 0\)
  3. \(m\frac{d^2\theta}{dt^2} + mg\cos\theta = 0\)
  4. \(m\frac{d^2\theta}{dt^2} - mg\cos\theta = 0\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of motion for a pendulum is given by (m\frac{d^2\theta}{dt^2} + mg\sin\theta = 0), where (m) is the mass of the pendulum bob, (g) is the acceleration due to gravity, (L) is the length of the pendulum, and (\theta) is the angle the pendulum makes with the vertical. This equation is a second-order nonlinear differential equation.

Multiple choice

Which of the following is an example of a chaotic system?

  1. Pendulum

  2. Double Pendulum

  3. Logistic Map

  4. Harmonic Oscillator
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The Logistic Map is a well-known example of a chaotic system. It is a simple mathematical equation that exhibits chaotic behavior, meaning it is highly sensitive to initial conditions and can produce unpredictable outcomes.

Multiple choice

Which of the following is an example of a strange attractor?

  1. Lorenz Attractor

  2. Hénon Attractor

  3. Rossler Attractor

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

The Lorenz Attractor is a famous example of a strange attractor. It is a chaotic attractor that exhibits a complex and unpredictable behavior, characterized by its butterfly-like shape.

Multiple choice

Which of the following is an example of a chaotic system that exhibits sensitive dependence on initial conditions?

  1. Pendulum

  2. Double Pendulum

  3. Logistic Map

  4. Harmonic Oscillator
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The Logistic Map is a well-known example of a chaotic system that exhibits sensitive dependence on initial conditions. It is a simple mathematical equation that can produce unpredictable outcomes due to its high sensitivity to initial values.

Multiple choice

What is the relationship between the pulsation period of a star and its mass?

  1. Pulsation period is proportional to mass

  2. Pulsation period is inversely proportional to mass

  3. Pulsation period is independent of mass

  4. Pulsation period is proportional to the square of mass
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The pulsation period of a star is inversely proportional to its mass. This is because more massive stars have stronger gravitational forces, which tend to suppress pulsations.