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Q14.23:
A circular disc of mass 10 kg is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be 1.5 s. The radius of the disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional spring constant $\alpha$ is defined by the relation $J = –\alpha \theta$, where $J$ is the restoring couple and $\theta$ the angle of twist).
Solution :
Step 1: Given Variables & Theoretical Foundation
To determine the torsional spring constant $\alpha$, we utilize the dynamics of a torsional pendulum. The system consists of a circular disc suspended by a wire, undergoing angular simple harmonic motion.
- Mass of the disc ($M$): $10\text{ kg}$
- Radius of the disc ($R$): $15\text{ cm} = 0.15\text{ m}$
- Period of oscillation ($T$): $1.5\text{ s}$
- Restoring torque relation: $J = -\alpha \theta$
[By the Principle of Angular Harmonic Motion, the period $T$ is given by $T = 2\pi \sqrt{\frac{I}{\alpha}}$, where $I$ is the Moment of Inertia of the disc about the axis of rotation.]
Step 2: Calculation of Moment of Inertia ($I$)
For a circular disc rotating about an axis passing through its centre and perpendicular to its plane, the moment of inertia is defined as:
$I = \frac{1}{2} M R^2$
Substituting the given values:
$I = \frac{1}{2} \times 10\text{ kg} \times (0.15\text{ m})^2$
$I = 5 \times 0.0225 = 0.1125\text{ kg}\cdot\text{m}^2$
Step 3: Deriving the Torsional Spring Constant ($\alpha$)
Rearranging the period formula $T = 2\pi \sqrt{\frac{I}{\alpha}}$ to solve for $\alpha$:
$\frac{T}{2\pi} = \sqrt{\frac{I}{\alpha}}$
$\frac{T^2}{4\pi^2} = \frac{I}{\alpha}$
$\alpha = \frac{4\pi^2 I}{T^2}$
Step 4: Final Substitution & Numerical Computation
Using $\pi^2 \approx 9.8696$:
$\alpha = \frac{4 \times 9.8696 \times 0.1125}{(1.5)^2}$
$\alpha = \frac{4.44132}{2.25}$
$\alpha \approx 1.9739\text{ N}\cdot\text{m/rad}$
Final Solution: The torsional spring constant of the wire is approximately $1.97\text{ N}\cdot\text{m/rad}$.
More Questions from Class 11 Physics Oscillations Additional Exercises
- Q14.20: An air chamber of volume $V$ has a neck area of cross section $a$ into which a ball of mass $m$ just fits and can move up and down without any friction (Fig.14.27). Show that when the ball is pressed down a little and released , it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal [see Fig. 14.27].
- Q14.21(a): You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (a) the spring constant $k$ and
- Q14.21(b): You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (b) the damping constant $b$ for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.
- Q14.22: Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.
- Q14.24(a): A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (a) 5 cm
- Q14.24(b): A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (b) 3 cm
- Q14.24(c): A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (c) 0 cm.
- Q14.25: A mass attached to a spring is free to oscillate, with angular velocity $\omega$, in a horizontal plane without friction or damping. It is pulled to a distance $x_0$ and pushed towards the centre with a velocity $v_0$ at time $t = 0$. Determine the amplitude of the resulting oscillations in terms of the parameters $\omega$, $x_0$ and $v_0$. [Hint : Start with the equation $x = a cos (\omega t+\theta)$ and note that the initial velocity is negative.]
CBSE Solutions for Class 11 Physics Oscillations
Chapters in CBSE - Class 11 Physics
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