A 110 kg mass on a spring has displacement as a function of

A 1.10 kg mass on a spring has displacement as a function of time given by the equation x(t)=(7.40cm)cos[(4.16rad/s)t2.42rad].

E.) Find the position of the mass at t=1.00s;

F.) Find the speed of the mass at t=1.00s;

G.) Find the magnitude of acceleration of the mass at t=1.00s;

H.) Find the magnitude of force on the mass at t=1.00s;

Solution

The motion of the spring is of the form

X(t) = ACos(t+)

A= 7.4 cm , = 4.16 rads/s , = -2.42 rads

Mass m = 1.10 kg

Find the time for one complete vibration.

Period of vibration T = 2/ = 1.51 s

Find the force constant of the spring.

= (k/m) = 4.16

spring constant k = (4.16)² x1.1 = 19.04 N/m

Find the maximum speed of the mass.

Velocity v = dx/dt = ASin(t+)

Maximum speed = A = 0.074*4.16 =0.3078 m/s

Find the maximum magnitude of force on the mass.

Maximum Force F = A² = 7.4x(4.16)² = 128.06 N

Find the position of the mass at t=1.00s;

X(t=1)=(7.40cm)cos[4.162.42]. = -1.25 cm

Find the speed of the mass at t=1.00s;

V(t=1) = ASin(t+) = 7.4*4.16Sin(4.16*1-2.42)

                          =30.34 cm/s = 0.3034 m/s

Find the magnitude of acceleration of the mass at t=1.00s;

Acceleration a = dv/dt = -A² Cos(t+)

                      A(t=1)    = 7.4*(4.16)² Cos(4.16-2.42)

                                    = 21.57 cm/s²    = 0.2157m/s²

Find the magnitude of force on the mass at t=1.00s;

F = ma = 1.1*0.2157 = 0.2372 N