(a) Sketch a diagram of a simple pendulum performing simple harmonic motion and indicate positions of maximum potential energy and kinetic energy.
(b) A body moving with simple harmonic motion in a straight line has velocity, v and acceleration, a, when the instantaneous displacement, x in cm, from its maximum position is given by x = 2.5 sin 0.4 (pi t), where t is in seconds. Determine the magnitude of the maximum (i) veloxity; (ii) acceleration
(c) A mass m attached to a light spiral is caused to perform simple harmonic motion of frequency
f = (frac{1}{2 pi} sqrt{frac{k}{m}}), where k is the force constant of the spring.
(i) Explain the physical significance of (sqrt{frac{k}{m}}).
(ii) If m = 0.30 kg, k = 30Nm(^{-1}) and the maximum position is 0.015m, calculate the maximum;
(i) kinetic energy
(ii) tension in the spring during the motion [g = 10 ms(^{-1}), (pi) = 3.142]
Explanation
(a)
(b)(i) V = rw = 2.5 x 10(^{-2}) x 0.4 (pi)
= 3.142 x 10(^{-2})m/s
(ii) a = rw(^2) = 2.5 x 1.0(^{-2}) x (0.4 (pi^2))
(c) (i) (frac{k}{m}) = w. H is the rate of change of angular displacement. H is measured in radians per second
(ii) (1) K.E(_{text{max}} frac{1}{2}MV^2_{text{max}} = frac{1}{2} mr^2w^2)
But w = (sqrt{frac{k}{m}} = w^2 = frac{k}{m} = frac{30}{0.3} = 100 rad/s^2)
K.E(_{text{max}} = frac{1}{2}mr^2w^2 = frac{1}{2} times 0.30 times (0.015)^2) x 100
= 3.38 x 10(^{3})J
T(_{text{max}}) = Weight + Ke = Mg + Ke
= 0.30 x 10 + 30 x 0.015
= 3.45N