A 1kg block is attached to a horizontal spring on a frictionless surface. It is

Question

A 1kg block is attached to a horizontal spring on a frictionless surface. It is found that it takes a force of 10 N to compress the spring 3 cm from equilibrium. The spring and block is then stretched to x = 4 cm from equilibrium and released from rest. a) Find the period and frequency of the motion. b) Write the position, velocity, and acceleration as functions of time. c) What are the magnitudes of the max speed and max acceleration? d) Suppose, instead of being released from rest, the block was given an initial velocity of -0.1 m/s (towards the equilibrium position). What would be the amplitude and phase constant for this case?

Explanation / Answer

F = k x
10 = k (0.03)

k = 333.33 N/m

(A) m w^2 = k

w = sqrt(k/m) = sqrt(333.33 / 1) = 18.3 rad/s

time period = 2 pi / w = 0.344 sec ......Ans

f = 1/T = 2.91 Hz .....Ans

(b) x = A cos(wt)

x = 4 cm cos(18.3t) .....Ans

v = dx/dt = - 0.732m/s sin(18.3t) ......Ans

a = dv/dt = - 13.4 m/s^2 cos(18.3t) .....Ans

(c) Vmax = 0.732 m/s

a_max = 13.4 m/s^2

(d) k A^2 /2 = k x^2 / 2 + m v^2 / 2

333.33 A^2 = (333.33)(0.04^2) + (1) (0.1^2)

A = 0.0404 m Or 4.04 cm .......Ans

4 = 4.04 cos( phi)

and - 0.1 = -0.732 sin(phi)

phi = 0.14 rad ........Ans

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