Name: Velocity and Acceleration in SHM
Uploaded: 2019-07-19
Duration: 5 min 47 s
Description: In this video, we will derive the equations and graphs of the velocity and acceleration of an object in SHM. Let us watch this video now.

Question

The figure shows a physical pendulum consisting of a uniform disk with radius $r = 10.0 cm$ and mass $500 g$ attached to a uniform rod with the length, $L = 500 mm$ and mass $250 g$ .

$(a)$ Calculate the rotational inertia of the pendulum about the pivot point/hinge.

$(b)$ What is the distance between the pivot point and the centre of mass of the pendulum?

$(c)$ Calculate the period of oscillation for small amplitude oscillations.

Accepted Answer

Given,

Radius of disk, $r = 10 cm = 0.1 m$

Mass of disk, $M = 500 g = 0.5 kg$

Length of rod, $L = 500 mm = 0.5 m$

Mass of the rod, $m = 250 gm = 0.25 kg$

$(a)$ Moment of inertia for a given system,

Moment of inertia of disk about its own axis, $I_{d} = \frac{M r^{2}}{2}$
Moment of inertia of disk about hinge using parallel axes theorem,
$I_{d, hinge} = \frac{M r^{2}}{2} + M {(r + L)}^{2}$

Moment of inertia of rod about the hinge, $I_{r} = \frac{m L^{2}}{3}$

The moment of inertia of combined system is

$I = \frac{M r^{2}}{2} + M {(r + L)}^{2} + \frac{m L^{2}}{3}$

$I = \frac{(0.5) {(0.1)}^{2}}{2} + 0.5 {(0.1 + 0.5)}^{2} + \frac{0.25 {(0.5)}^{2}}{3}$

$I = \frac{0.5 \times 0.01}{2} + 0.5 {(0.6)}^{2} + \frac{0.25 \times 0.25}{3}$

$I = 0.203 kg m^{2}$

$(b)$ Distance of centre of mass of the disk from hinge, $L_{D} = (L + r)$

$L_{D} = 0.6 m$

Distance of centre of mass of rod from hinge, $L_{R} = \frac{L}{2}$

$L_{R} = 0.25 m$

Distance of centre of mass of pendulum from hinge,

$h = \frac{M L_{D} + m L_{R}}{M + m}$

$h = \frac{(0.5 \times 0.6) + (0.25 \times 0.25)}{0.5 + 0.25}$

$h = 0.48 m$

$(c)$ Time-period of compound pendulum is given by,
$T = 2 π \sqrt{\frac{I}{(M + m) g h}}$

$T = 2 π \sqrt{\frac{0.203}{0.75 \times 9.8 \times 0.48}}$

$T = 2 π \times \sqrt{0.057}$

$T = 1.5 s$

Important Questions on Oscillations

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