A 0.97 kg block oscillates back and forth along a straight line on a frictionles

Question

A 0.97 kg block oscillates back and forth along a straight line on a frictionless horizontal surface. Its displacement from the origin is given by
x = (13 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

from the given wave equation

m = 0.97 kg

A = 13 cm = 0.13 m

w = 13 rad/s

(a)

we know that

w = 2*pi*f

so, f = w/2*pi = 13/2*3.14

f = 2.07 Hz

(b)

maximum speed v = A*w

v = 0.13*13 = 1.69 m/s

(c)

v will be maximum at x = 0

(d)

maximum acceleration a = A*w^2 = 0.13*13^3 = 21.97 m/s^2

(e)

as we know that

a = w^2*x

maximum acceleration accures at x = +/-A

at x = +/-13 cm

(f)

force F = -k*x

w = sqrt(k/m)

k = w^2*m = 13^2*0.97

k = 163.93 N/m

F = -163.93*.13

F = -21.31 N

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