Question

A mass of 100 g is attached to a spring and oscillating with simple harmonic motion. The position of the mass at all times is given by

x(t) = (2.0 cm) cos(2t),

where t is in seconds, and the 2 is in rad/s. Determine the following:

(a) The amplitude (in cm).
cm


(b) The frequency.
Hz


(c) The maximum speed in cm/s. Think about the expression you can write for v(t). Where is the maximum velocity in that expression? You can also use energy to find the maximum speed.



(d) The velocity at t = 2.51 s. HINT: Use the information you have found already to write out the expression for v(t). Then evaluate your expression at t = 2.51 s. NOTE that your answer should be in cm/s
cm/s

(e) In which direction is the mass moving at this time? ---Select---neither, it is at a turnaround pointin the negative x directionin the positive x direction

Answer 1

a) from the displacement equation

A= 2 cm

b) angular frequency from the displacement equation is,

=2 rad/s

f=2/(2*) = 0.3184 Hz

c) v= -A*sin(t) = -Asqrt(1-cos²(t)) {differentiating the displacement expression with respect to time}

v= -*sqrt(A²-x²)

now looking at the above expression we can say that velocity is maximum when x=0

v_max = -A = -2*2

v_max =-4 cm/s {negative sign tells us that the direction of velocity is opposite to that of displacement}

d)  v= -A*sin(t) = -2*2*sin(2*2.51)

v=3.8122 cm/s

e) since the velocity is positive hence it must be in the negative x direction