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Harmonic Oscillations

Q- A harmonic force F0 sin(ωt) is applied to mass m1 which in turn is coupled to springs of spring constants k1, k2 and k3. Let x1 and x2 ­be the deviations from equilibrium for m1 and m2, respectively.  The blocks moves on a horizontal frictionless table.

(a) Write the equations of motion for …

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Forced Damped Harmonic Oscillator

Q- A simple fatigue testing machine has a moving base that is 15kg and is constrained to only move vertically. The base is supported by a rubber block with a spring constant, k = 18 kN/m and a viscous damping constant of c = 200N.s/m. A rotating mass which is part of the base creates a harmonic forc…

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Simple Harmonic Motion

(On request-Raunak)

Q- Two particles executes SHM of same amplitude of 20 cm with same period along the same line and about same equilibrium position. If the maximum distance between them is 20 cm, find their phase difference.

Solution:

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SHM: Damped Oscillation

Q- A 1.5 kg piston supported on a helical spring vibrates freely with a natural frequency of 3Hz. When oscillating within an oil filled cylinder the frequency of free oscillation is reduced to 2.90Hz.

(a) Find the stiffness of the helical spring.

(b) Find the damping ratio when in the oil filled …

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Simple Harmonic Motion

Q- A particle executes linear harmonic motion about the point x = 0. At t = 0, it has a displacement x = 0.37 cm and zero velocity. The frequency of oscillation is 0.25 Hz. Determine

(a) The period (b) the angular frequency (c) the amplitude (d) the displacement at time t (e) the velocity at time t…

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Simple Harmonic Motion

Q- A block is suspended by a vertical spring of constant 1800 N/m and makes 5.5 oscillations per second.

(a) Find the mass of the block.

(b) What is elongations in the spring if the block is at rest in equilibrium?

(c) If the block is pulled down by 2.5 cm from equilibrium position and released, …

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Simple Harmonic Motion

Q- A particle M executes simple harmonic motion along x axis. The reference particle for this SHM moves on a circular path of radius R = 40 cm with a constant speed of 80 cm/s.

(a) Find the angular frequency, frequency and the time period for this SHM.

(b) If at t = 0, the reference particle makes…

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Rotational Simple Harmonic Motion.

Q- A very light rigid rod with a length of 0.500m extends straight out from one end of a meter-stick. The meter-stick is suspended from a pivot at the far end of the rod and is set into oscillation in a vertical plane. Determine the period of oscillation.

Solution:

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Simple Harmonic Motion

Q- A bungee jumper whose mass is 95 kg, has just completed his first bungee jump and is now bouncing up and down at the end of the cord. His oscillations have initial amplitude of 7 m and a period of 4.0 s.

(a) What is the spring constant of the bungee cord?

(b) What is the jumper's maximum speed …

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Simple Harmonic Motion

Q- A 240 g mass suspended by a 0.8 m long string is pulled 8.4° to one side and released. How long does it take for the pendulum to reach 6.0° on the opposite side? 

Solution:

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Simple Harmonic Motion

Q- (a) When the displacement of a mass on a spring is one-sixteenth of the amplitude A, what fraction of the energy is kinetic and what fraction is potential? 

(b) At what displacement, as a fraction of A, is the energy one-eighth kinetic and seven-eighths potential? 

Solution:

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Simple Harmonic Motion

Q- A small block of mass 80 g is oscillating on a spring. The equation of motion for its position x as a function of time is given by

                   x(t) = (5.0 cm) cos (10t + π/4)

(a) what is the amplitude of the oscillation?

(b) What is the period of oscillation?

(c) What is the…

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Simple Harmonic Motion

Q- Astronauts in space cannot weigh themselves by standing on a bathroom scale. Instead, they determine their mass by oscillating on a large spring. Suppose an astronaut attaches one end of a large spring to her belt and the other end to a hook on the wall of the space capsule. A fellow astronaut th…

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Simple Harmonic Motion

Q- 240 g mass on a 0.8 m long string is pulled 8.4° to one side from its equilibrium position and released. How long does it take for the pendulum to reach 6.0° on the opposite side?

Solution:

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Simple Harmonic Motion

Q. A 170 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has vx = -28 cm/s. Determine the following.

(a) The period  (b) The angular frequency  (c) The amplitude  (d) The phase constant  (e) The maximum speed  (f) The maximum acc…

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Simple Harmonic Motion

A block attached to a spring with unknown spring constant oscillates with a period of 3.0 s.

(a) What is the period if the mass is doubled?

(b) What is the period if the mass is halved?

(c) What is the period if the amplitude is doubled?

(d) What is the period if the spring constant is doubled?

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Simple Harmonic Motion

Q- A block with mass M attached to a horizontal spring with force constant k is moving with simple harmonic motion having an amplitude A . At the instant when the block passes through its equilibrium position, a lump of putty with mass m is dropped vertically onto the block from a very small height …

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Simple Harmonic Motion

Q- A piston in a gasoline engine is in simple harmonic motion. Taking the extremes of its position relative to its center point as plus or minus 5.00cm, find the maximum velocity and acceleration of the piston when the engine is running at the rate of 3600 rev/min.

Solution:

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Simple Harmonic Motion

Q- A block-spring system oscillates with amplitude of 3.50 cm. The spring constant is 250 N/m and the mass of the block is 0.600 kg.

(a) Determine the mechanical energy of the system.

(b) Determine the maximum speed of the block.

(c) Determine the maximum acceleration.

Solution:

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SHM

A uniform cylinder of mass m is rotated through a small angle from the equilibrium position and released. The stiffness of the spring fixed to the cylinder is k. The radius of the cylinder is r. The spring is connected at a distance a from the center of the cylinder. The cylinder rolls without slipp…

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