A 0.75 kg block oscillates back and forth along a straight line on a frictionles

Question

A 0.75 kg block oscillates back and forth along a straight line on a frictionless horizontal surface. Its displacement from the origin is given by x = (15 cm)cos[(13 rad/s)t + /2 rad] (a) What is the oscillation frequency? (b) What is the maximum speed acquired by the block? (c) At what value of x does this occur? (d) What is the magnitude of the maximum acceleration of the block? (e) At what positive value of x does this occur? (f) What force, applied to the block by the spring, results in the given oscillation?

Explanation / Answer

general equation of motion

x = a cos ( wt + b)

x = 15 cos (13t + pi/2)

Amplitude A = 15 cm

angualr frequncy W = 13 rad/s

part A :

W = 2pi f = 13

f = 13/2[i

f = 2.07 Hz

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part b :

Vmax = Alpha W

Vmax = 0.15 * 13

Vmax = 1.95 m/s

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part C:

this occures when sin (3 t + pi/2) = +-1

X = 13 cos ( +- pi/2)

X = 0

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part D:

max accleration alpha A = W^2 a

ALpha A = 13^2 * 0.15 = 25.35 rad/s^2

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part E :

cos (13 t + pi/2) = -1

x = -15 cm

F = - kx   

F = - MW^2 x

F = 0.75 * 13^2 * x

F = - 126.75 x

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