Question

In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression, x = 7.00 cos(3t + π/7) where x is in centimeters and t is in seconds.

(a) At t = 0, find the position of the piston. ____ cm

(b) At t = 0, find velocity of the piston. ____ cm/s

(c) At t = 0, find acceleration of the piston. ____ cm/s^2

(d) Find the period and amplitude of the motion.

period = ____ s

amplitude = ____ cm

Answer 1

Given expression is -

x(t) = 7.00 cos(3t + π/7)

Differentiate this with respect to t -

x'(t) = v(t) = -21.0 sin(3t + π/7)

Again differentiate -

x''(t) = a(t) = -63.0cos(3t + π/7)

where you need to differentiate using the chain rule

(a) At t = 0, position = x(0) = 7.0 cos(π/7) = 6.31 cm

(b) At t = 0, velocity = v(0) = -21.0 sin(π/7) = -9.11 cm/s

(c) At t = 0, acceleration = a(0) = -63.0 cos(π/7) = -56.76 cm/s²

Now, comparing 7.0cos(3t + π/7) with Acos(wt + φ)
where A is the amplitude, w = 2π/T is the angular frequency, T is the period, and φ is the phase shift
you get

(d) Period (T) -  
w = 2π/T = 3.0

=> T = 2π/3.0 = 2.09 s

(e) Amplitude, A = 7.0 cm