# Numerical Problems : Simple Harmonic Motion

#### LEVEL – I

Q:1. The equation of a simple harmonic motion is given by x = 6 sin 10 t + 8 cos 10 t, where x is in cm, and t is in seconds. Find the resultant amplitude.

Ans: 10 cm

Q:2. A particle of mass 4 g performs S.H.M. between x = – 10 cm and x = + 10 cm along x-axis with frequency 60 Hz, initially the particle starts from x = +5 cm. Find

(a) equation of motion of the particle.

(b) the initial phase, amplitude and time period of the particle.

(c) velocity- displacement and acceleration displacement curve of this motion.

(d) plot the graphs of (i) KE vs displacement

(ii) PE vs displacement

(iii) Total energy vs displacement

Ans: (a)x = 10 sin(120 πt + π/6)
(b) π/6 , 10 cm , 1/60 sec
(c) $\displaystyle \frac{v^2}{\omega^2 A^2} + \frac{x^2}{A^2} = 1$ ; Which is an equation of ellipse ,
acceleration of the particle is given by a = -ω2 x

Q:3. A cubical body (side .1 m and mass 0. 02 kg) floats in water. It is pressed and then released so that it oscillates vertically. Find the time period. (density of water = 1000 kg/m3).

Ans: 0.0885 sec

Q:4. Find the time period of the motion of a particle shown in figure. Neglecting the small effect of the bend near the bottom.

Ans: 0.726 sec

Q:5. Consider a situation shown in the figure. Show that if the blocks are displaced slightly in opposite directions and released, they will execute S.H.M. calculate the time period.

Ans: $\displaystyle \omega = \sqrt{\frac{k}{\mu}} \; , \mu = \frac{m_1 m_2}{m_1 + m_2} \; , \; m_1 = 2m \; , \; m_2 = 3m$

Q:6. A uniform rod of mass m and length l is pivoted at one end. It is free to rotate in a vertical plane. Find the time period of oscillation of rod if it is slightly displaced from vertical and released.

Ans: $\displaystyle T = 2\pi\sqrt{\frac{2 l}{3 g}}$

Q:7. A particle is executing SHM. A and B are the two points at which its velocity is zero. It passes through a certain point P at intervals of 0.5 and 1.5 sec with a speed of 3 m /s. Determine the maximum speed and also the ratio AP/BP.

Ans: 3√2 m/s ; $\displaystyle \frac{AP}{BP} = \frac{\sqrt{2}-1}{\sqrt{2}+1}$

Q:8. A ball is suspended by a thread of length L at the point O on the wall PQ which is inclined to the vertical by a small angle α to the thread with the ball is now displaced through a small angle β away from the vertical and also from the wall if the ball is released, find the period of oscillation of the pendulum when

(a) β < α

(b) β > α    Assume the collision on the wall to be perfectly elastic.

Ans : (a)$\displaystyle T_1 = 2\pi \sqrt{\frac{L}{g}}$

(b)$\displaystyle T_2 = \sqrt{\frac{L}{g}} [\pi + 2sin^{-1}(\alpha/\beta)]$ ; Obviously which is less than T1

Q:9. A small solid cylinder of mass M attached to a horizontal massless spring can roll without slipping along a horizontal surface. Show that if the cylinder is displaced and released, if executes S.H.M. Also find its time period.

Ans: $\displaystyle T = 2\pi \sqrt{\frac{3M}{2k}}$

Q:10. The friction coefficient between the two blocks shown in figure is μ and the horizontal plane is smooth (a) If the system is slightly displaced and released find the magnitude of the frictional force between the blocks when the displacement from the mean position is x. (b) what can be the maximum amplitude if the upper block does not slip relative to the lower block ?

Ans: $\displaystyle \mu m\frac{(M+m)g}{M k}$