A mass is attached to the end of a spring and set into oscillation on a horizont

Question

A mass is attached to the end of a spring and set into oscillation on a horizontal frictionless surface by releasing it from a compressed position. The record of time is started (i.e., we'll define t = 0) when the oscillating mass first passes through the equilibrium position, and the position of the mass at any time is described by

x(t) = (4.1 cm)sin[(9.3 rad/s)?t].

Notice that the expression in this problem for x(t) is written as a sine function rather than a cosine function. This is OK. It just means that the oscillation starts with x = 0 when t. Determine the following:

(a) frequency of the motion
Hz

(b) period of the motion
s

(c) amplitude of the motion
cm

(d) first time after

t = 0

that the object reaches the position

x = 2.6 cm.

Note that in this case, you can't do any "tricks" to find the time; i.e. it won't be some simple fraction of the period. You will need to set x(t) = 2.6 cm and then solve for t.
s

Explanation / Answer

Given that x(t) = (4.1 cm)sin[(9.3 rad/s)?t].

The above equation is in the form of x(t) =Asinwt

The frequency of the motion is given by

w =9.3rad/sPi

2pif =9.3rad/sPi then 2f =9.3rad/s

Frequency f =4.65Hz

Now the period of the motion is given by f=1/T then T =1/f =1/4.65 =0.215s

The amplitude of the motion is A = 4.1cm

If x(t) =2.6cm to find time

             2.6cm =4.1cmsin(9.3rad/s)pit

            0.634 =sin(9.3rad/s)pit)

            9.3rad/s)pit =sin^-1(0.634) =39.34

                              time (t) =39.34/9.3*3.14 =1.34s

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