Question

1. Consider an object on a spring of k=200 N/m whose position is given by x=(5.00 cm)cos(9.90 rad/s). Find: (a) the maximum speed of the object and when this maximum speed first occurs after t=0, (b) the maximum of the acceleration of the object and when the maximum of the magnitude of the acceleration first occurs after t=0, (49.5cm/s 490 cm.s^2), (c) the amplitude and period of motion, (d) the position of the block at t=1.0s, (d) the maximum kinetic energy and potential energy of the spring block system.

Answer 1

The motion of the object can be described by SHM like

x(t) = 5.0 Cos(t ) cm

= 9.9 rads/s - angular speed

speed of the object u = dx/dt = -xSin(t)

The maximum speed is when Sin(t) =1 and it is for t > 0

max speed = |x| = 5.0*9.9 = 49.5 cm/s

acceleration - d²x/dt² = -²*5.5 Cos(t ) cm/s/s

max. acceleration=|-² *x| =  5.0*9.9² = 490.05 cm/s/s

c) maximum amplitude A = 5.0 cm

period of motion T = 2/ = 0.635 s

d) pos. at t=1s x= 5.0*Cos(9.9) = -4.45 cm

e) max. pot. energy and KE of the block

= 1/2 kA² 1/2 *200 * 0.05² = 0.25 J