A mass on a spring vibrates horizontally

Question

A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is x(t) = 4 cos(t),
where t is in seconds and x is in centimeters. Find the position, velocity, and acceleration of the mass at time t = 5pi/6.

 

 

Answer 1

Position, velocity and acceleration are all based on the equation of motion.

The equation represents the position.  The first derivative gives the velocity and the 2nd derivative gives the acceleration.

 

Given x(t) = 4 cos(t), we can differentiate

x'(t) = -4 sin (t)

x"(t) = -4 cos (t)

 

At t=5π/6

 

Position, x(t) =  4 cos (5π/6) = -4(√3)/2 = -2(√3) cm

Velocity, x'(t) = -4 sin (5π/6) = -4(1/2) = -2 cm/s

Acceleration x"(t) = -4 cos (5π/6) = 4(√3)/2 = 2(√3) cm/s²

 

Hope this helps.