A 50 gm mass at the end of a spring vibrates in SHM. The amplitude of the motion is 12 c.m. and the period is 1.70 s. Find (a) the frequency (b) the spring constant (c) the maximum speed of the mass (d) the maximum acceleration of the mass (e) the speed when the displacement is 6 c.m. (f) the acceleration when x = 6 c.m.

Explanation

(a) Frequency `f = (1)/(T) rArr (1)/(1.70 s) = 0.500 Hz`
(b) Since `T = 2pi sqrt((m)/(k)), " "K = (4pi^(2)m)/(T^(2))`
`K = (4pi^(2)(0.050 kg))/((1.70 s)^(2)), " "K = 0.68 N//m`
(c) Maximum speed of the mass
`v_(max) = A omega = (0.12 m). (2pi)/(1.75) , v_(max) = 0.44 m//s`
(d) Acceleration of the body
`a = - omega^(2)y`
Max. acceleration
`a_(max) =- omega^(2)A`
`a_(max) =-((2pi)/(T))^(2).A=-((2pi)^(2))/((1.70 s)^(2)) xx 0.12`
`a_(max) = 1.63 m//s^(2)`
(e) Speed of the body when the displacement is 6 cm
`v = omega sqrt(A^(2) - y^(2)) = (2pi)/(T) sqrt(A^(2) - y^(2))`
`= (2pi)/(1.70) sqrt((0.12)^(2) - (0.06)^(2)) = 0.38 m//s`
(f) Acceleration of body when y = 6 c.m.
`a = - omega^(2)y =- ((2pi)/(T))^(2).y =-((2pi)^(2))/((1.70)^(2))xx 0.06`
`a =- 0.82 m.s^(2)`