Question
A thin solid disk of 1 kg is rotating along its diameter axis at the speed of 1800 rpm . By applying an external torque of $25\ \pi \mathrm\ {Nm}$ for 40s , the speed increases to 2100 rpm . The diameter of the disk is _______________ m.
✓
Answer
(40)
Given, $\mathrm{m}=1 \mathrm{~kg}$
$\omega_{\mathrm{i}}=1800 \mathrm{rpm}=1800 \times \frac{2 \pi}{60}=60 \pi \frac{\mathrm{rad}}{\mathrm{sec}}$
$\omega_{\mathrm{f}}=2100 \mathrm{rpm}=2100 \times \frac{2 \pi}{60}=70 \pi \frac{\mathrm{rad}}{\mathrm{sec}}$
$\tau_{\text {ext }}=25 \pi \mathrm{Nm}$
$\mathrm{t}=40 \mathrm{sec}$
Using equation of motion
$\omega_{\mathrm{f}}=\omega_{\mathrm{i}}+\alpha \mathrm{t}$
$70 \pi=60 \pi+\alpha(40)$
$\alpha=\frac{\pi}{4} \mathrm{rad} / \mathrm{sec}^{2}$
Also, $\tau=\mathrm{I} \alpha$
$\tau=\frac{\mathrm{mR}^{2}}{4} \alpha$
$25 \pi=\frac{1 \times \mathrm{R}^{2}}{4} \times \frac{\pi}{4}$
$\mathrm{R}=20 \mathrm{~m}$
Hence, diameter of disk $=2 \mathrm{R}=2 \times 20=40 \mathrm{~m}$
Given, $\mathrm{m}=1 \mathrm{~kg}$
$\omega_{\mathrm{i}}=1800 \mathrm{rpm}=1800 \times \frac{2 \pi}{60}=60 \pi \frac{\mathrm{rad}}{\mathrm{sec}}$
$\omega_{\mathrm{f}}=2100 \mathrm{rpm}=2100 \times \frac{2 \pi}{60}=70 \pi \frac{\mathrm{rad}}{\mathrm{sec}}$
$\tau_{\text {ext }}=25 \pi \mathrm{Nm}$
$\mathrm{t}=40 \mathrm{sec}$
Using equation of motion
$\omega_{\mathrm{f}}=\omega_{\mathrm{i}}+\alpha \mathrm{t}$
$70 \pi=60 \pi+\alpha(40)$
$\alpha=\frac{\pi}{4} \mathrm{rad} / \mathrm{sec}^{2}$
Also, $\tau=\mathrm{I} \alpha$
$\tau=\frac{\mathrm{mR}^{2}}{4} \alpha$
$25 \pi=\frac{1 \times \mathrm{R}^{2}}{4} \times \frac{\pi}{4}$
$\mathrm{R}=20 \mathrm{~m}$
Hence, diameter of disk $=2 \mathrm{R}=2 \times 20=40 \mathrm{~m}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Three waves of equal frequency having amplitudes $10 \,\mu \, m, 4 \, \mu \,m$ and $7 \mu m$ arrive at a given point with successive phase difference of $\frac{\pi }{2}$. The amplitude of the resulting wave in $\mu \,m$ is given by
→A liquid at $30^{\circ} C$ is poured very slowly into a Calorimeter that is at temperature of $110^{\circ} C$. The boiling temperature of the liquid is $80^{\circ} C$. It is found that the first $5 gm$ of the liquid completely evaporates. After pouring another $80 gm$ of the liquid the equilibrium temperature is found to be $50^{\circ} C$. The ratio of the Latent heat of the liquid to its specific heat will be. . . . .${ }^{\circ} C$. [Neglect the heat exchange with surrounding]
→Consider the diagram shown below. A voltmeter of resistance $150\,\Omega$ is connected across $A$ and $B$. The potential drop across $B$ and $C$ measured by voltmeter is $...........\,V$
→A fist looking from within water sees the outside world through a circular horizon. If the fish $\sqrt 7 cm$ below the surface of water, what will be the radius of the circular horizon.....$cm$
→A travelling wave pulse is given by $y=\frac{4}{3 x^2+48 t^2+24 x t+2}$ where $x$ and $y$ are in metre and $t$ is in second. The velocity of wave is ........... $m / s$
→A $9\,V$ stabilised voltage supply is required to run a car stereo system from the car's $12\,V$ battery. A zener diode with $V_z = 9\,V$ and $P_{max} = 0.27\, W$ is to be used as the voltage regulator. The load resistance is $450\,\Omega $. The value of series resistor is....$\Omega $
→If the equation of transverse wave is $y = 5\sin 2\pi \left[ {\frac{t}{{0.04}} - \frac{x}{{40}}} \right]$, where distance is in $ cm$ and time in second, then the wavelength of the wave is .... $cm$
→A beam of light converges at a point $P$. Now a lens is placed in the path of the convergent beam $12\, cm$ from $P$. At what point does the beam converge if the lens is a concave lens of focal length $16\,cm$ ?.......$cm$
→Net resistance of the given circuit between $X$ and $Y$ is ............... $\Omega$
→A capillary tube of radius $0.2\,cm$ is dipped vertically in a beaker containing liquid. If the liquid rises to a height of $5\,cm$ for which the angle of contact is $60^o$ , then surface tension of the liquid is ...... $dynes/cm$ $(d = 1\,gm/cm^3)$
→