Restoring force

Select the correct expression for a restoring force for a system moving in simple harmonic motion.

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The restoring force is trying to return an oscillating object to its centre position. This force is proportional to the distance from that centre position.

Pendulum

A pendulum is released from a maximum displacement, which is 10 cm from the centre of its swing. It returns to its original position in 2s. Calculate the position of the pendulum after 10 seconds from release.

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\(w= 2\pi/T\approx3.14rad/s\)

The pendulum performs periodical motion, so its position can be calculated using the following equation: \(x=Acos(wt)= 0.1cos(3.14\times 10)\approx0.1m\)

Period of a pendulum

Select the correct expression for the period of a pendulum with a length of l m and a mass of 10kg. 

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\(T = 2\pi \sqrt{l\over g}\)

The period of a pendulum is only dependent on the length of the pendulum string, and g on the planet on which it has been set up.

Velocity in SHM.

An object undergoes simple harmonic motion, its position is a sinusoidal function of time: \(x=Acos(wt)\). Select the correct expression for the velocity of this object.

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If \(x=Acos(wt)\), then the velocity of an object is \(v={dx\over dt}=Awcos(wt)\).

Acceleration in SHM

 An object undergoes simple harmonic motion, its position is a sinusoidal function of time: \(x=Asin(wt)\). Select the correct expression for the acceleration of this object.

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\(x = Asin(wt)\)

The velocity of an object: \(v ={dx\over dt}=Awcos(wt)\)

The acceleration of an object: \(a ={dv\over dt}=-Aw^2sin(wt)\).

Energy in SHM

A pendulum with a mass of 10 kg is raised 4m above its position of equilibrium. Calculate its maximum kinetic energy during the period of oscillation.

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Using the law of conservation of energy, \(\Delta E_k+\Delta E_p=0\). In a point with maximum elevation the kinetic energy equals zero and the potential energy has its largest value. In the position of equilibrium the kinetic energy has its largest value and the potential energy is minimal. 

So, \(\Delta E_k+mg(0-h)=0\Leftrightarrow {E_k}_{max}=mgh\approx 400J\)

Natural frequency

Select the correct definition of natural frequency of an oscillating system.

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Natural frequency is the frequency at which a system tends to oscillate in the absence of any driving or damping force. If forced frequency is equal to the natural frequency, the amplitude of vibration increases manyfold. This phenomenon is known as resonance.

Resonance

Select the correct definition of resonance into an oscillating system.

 

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Resonance is the phenomenon that is observed when an oscillating system is driven at its natural frequency so that it oscillates with a large amplitude.

Undamped oscillations

The displacement-time plot for an oscillating system is shown on a picture below. Which of the curves shows undamped oscillations?

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Undamped oscillations are observed in the case when there is no significant damping in the system.

Underdamping is observed when the system completes several oscillations, whose amplitude decreases exponentially.

Overdamping is observed when the system cannot complete even one cycle of oscillations.

The system is critically damped, if the damping is such that the oscillator returns to its equilibrium position in the quickest possible time, without going past that position.

Overdamping

The displacement-time plot for an oscillating system is shown on a picture below. Which of the curves shows overdamping?

Select the correct answer.

 

Undamped oscillations are observed in the case when there is no significant damping in the system.

Underdamping is observed when the system completes several oscillations, whose amplitude decreases exponentially.

Overdamping is observed when the system cannot complete even one cycle of oscillations.

The system is critically damped, if the damping is such that the oscillator returns to its equilibrium position in the quickest possible time, without going past that position.

Critically damped oscillator

The displacement-time plot for an oscillating system is shown on a picture below. Which curve shows the behavior of a critically damped oscillator?

Select the correct answer.

 

Undamped oscillations are observed in the case when there is no significant damping in the system.

Underdamping is observed when the system completes several oscillations, whose amplitude decreases exponentially.

Overdamping is observed when the system cannot complete even one cycle of oscillations.

The system is critically damped, if the damping is such that the oscillator returns to its equilibrium position in the quickest possible time, without going past that position.

Underdamped oscillator

The displacement-time plot for an oscillating system is shown on a picture below. Which of the curves shows the behavior of an underdamped oscillator?

Select the correct answer.

 

Undamped oscillations are observed in the case when there is no significant damping in the system.

Underdamping is observed when the system completes several oscillations, whose amplitude decreases exponentially.

Overdamping is observed when the system cannot complete even one cycle of oscillations.

The system is critically damped, if the damping is such that the oscillator returns to its equilibrium position in the quickest possible time, without going past that position.