A mass on a spring vibrates horizontally on a smooth level surface. its equation of motion is x (t) = 8 sin (t), where t is in seconds and x in centimeters. find the velocity and acceleration of time. find the position, velocity, and acceleration of the mass at time t=2pi/3. in what direction is it moving at that time

Solution:

we have given the equation of motion is x (t) = 8sint [where t in seconds and x in centimeter]

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.

x (t) = 8sint

x' (t) = 8cost

x" (t) = -8sint

now at time t=2pi/3,

position, x (t) = 8sin (2pi/3) = 4*squart (3) cm.

velocity, x' (t) = 8cos (2pi/3) = =4cm/s

acceleration, x" (t) = =8sin (2pi/3) = -4cm/s^2

so at present the direction is in y-axis.