Paper Questions (Raw)
Q1. A particle of mass (m=5 kg) is moving with a uniform speed (v=3 \sqrt{2} m/s) in the (x-y) plane along the line (y=x+4). The magnitude of the angular momentum, in SI unit, about origin is
1) zero
2) (40 \sqrt{2}) unit
3) 7.5 unit
4) 60 unit
Hint: (r_\perp=4 \sin 45^{\circ}=2 \sqrt{2}) unit$$L=mvr_\perp=5 \times(3 \sqrt{2}) \times(2 \sqrt{2})=60$$ unit
Q2. A solid sphere is rotating in a free space. If the radius of the sphere is increased, keeping the mass same, which one of the following will decrease?
1) Rotational kinetic energy
2) Moment of inertia
3) Angular momentum
4) Torque
Hint: R increases (I \propto R^2 \Rightarrow) I increases (L=I \omega \Rightarrow L) increases (\tau=I \alpha \Rightarrow \tau) increases In free space, (\tau=0 \Rightarrow L=) constant (E=\frac{L^2}{2 I} \Rightarrow E \propto \frac{1}{I} \propto \frac{1}{R^2} \Rightarrow E) decreases
Q3. A stone of mass m , tied to the end of a string, is whirled around in a horizontal circle. (Neglect the force due to gravity). The length of the string is reduced gradually, keeping the angular momentum of the stone about the centre of the circle constant. If the tension in the string is given by (T=Ar^n), where A is a constant, r is the instantaneous radius of the circle, then the value of n is
Hint: (L=mvr=) constant
(vr=c \Rightarrow v=\frac{c}{r}) $$T = \frac{mv^2}{r} \\ = \frac{m}{r}(\frac{c}{r})^2 = \frac{mc^2}{r^3}$$
(T \propto r^{-3}) (\therefore n=-3)
Q4. A ballet dancer, spinning with kinetic energy E, pulls her arms close to the body, thereby changing her M.I. by 50%. The kinetic energy now becomes
1) 0.67 E
2) E
3) 0.33 E
4) 2E
Hint: (E=\frac{L^2}{2 I} \Rightarrow E \propto \frac{1}{I}) (\frac{E_2}{E_1}=\frac{I_1}{I_2}) As dancer brings her arms close to her body, her M.I. decreases by 50%. (\therefore I_2=0.5 I_1) (\therefore \frac{I_1}{I_2}=2) (\therefore E_2=2 E_1)
Q5. A disc of M.I. I, rotating with angular velocity (\omega), is coupled with another stationary disc of M.I. 2I. The two attain a constant angular velocity after a short interval ' (t) '. The torque acting on the stationary disc is
1) (\frac{I \omega}{t})
2) (\frac{2 I \omega}{3 t})
3) (\frac{I \omega}{6 t})
4) 0
Hint: (\omega'=\frac{I \omega+2 I(0)}{I+2 I}=\frac{I \omega}{3 I}=\frac{\omega}{3}) (\tau=I \alpha=\frac{2 I(\frac{\omega}{3}-0)}{t}=\frac{2 I \omega}{3 t})
Q6. A person sitting firmly over a rotating stool has his arms stretched. If he folds his arms, his kinetic energy about the axis of rotation
1) increases.
2) decreases.
3) remains unchanged.
4) doubles.
Hint: (E=\frac{L^2}{2 I}) As I decreases, E increases.
Q7. A wheel of M.I. (2 kg-m^2) and rotating with a speed of (30 rev/sec) is coupled with another wheel of M.I. (3 kg-m^2) and at rest. The resultant speed of rotation is
1) (12 \pi) rad/s
2) (24 \pi) rad/s
3) (25 \pi) rad/s
4) (27 \pi) rad/s
Hint: (I_1 n_1+I_2 n_2=(I_1+I_2) n) ([2 \times 30]+[3 \times 0]=(2+3) n) (n=\frac{60}{5}=12) (\omega=2 \pi n=2 \pi \times 12=24 \pi)
Q8. When torque acting upon a system is zero, which of the following will be zero?
1) Change in angular momentum
2) Work done
3) Angular momentum
4) Rate of change of kinetic energy
Hint: From the relation (\tau = dL), if torque is zero, the change in angular momentum is zero.
Q9. Unit of angular momentum is
Q10. When viewed in direction of negative Z axis, the angular momentum of a body, moving in circular path, in clockwise direction in, xy plane, will be along
1) +z axis
2) - z axis
3) +y axis
4) - y axis
Q11. Two wheels of M.I., (2 kg-m^2) and (4 kg-m^2) rotate with angular velocity of (120 r.p.m.) and (60 r.p.m.) respectively in the same direction. If the two wheels are coupled so as to rotate with the same angular velocity, the KE of rotation of the coupled system is approximately
1) 126 J
2) 192 J
3) 279 J
4) 426 J
Hint: $$n = \frac{I_1 n_1+I_2 n_2}{I_1+I_2}=\frac{2(120)+4(60)}{2+4} \\ = \frac{240+240}{6}=80 rpm=\frac{8}{6} rps$$
(KE=\frac{1}{2}(I_1+I_2) \omega^2) $$ = \frac{1}{2} \times(2+4) \times(2 \pi \frac{80}{60})^2 \\ =3 \times(64)=192 J$$
Q12. A body of mass 10 kg , moves along a line (y=3) with speed of (10 m/s). Its angular momentum, about origin, is
1) 0
2) (90 kgm^2/s)
3) (150 kgm^2/s)
4) (300 kgm^2/s)
Hint: (L=mvr=10 \times 10 \times 3=300 kgm^2/s)
Q13. The frequency of rotation of a body, having constant angular momentum of L, change s 3n to 5n, in time (t). The change in K.E. of body is
1) (2 \pi nL)
2) (4 \pi nL)
3) (5 \pi nL)
4) nL
Hint: (K=\frac{1}{2} L \omega) (\Delta K=\frac{1}{2} L(\Delta \omega)=\frac{1}{2} L(2 \pi n_2-2 \pi n_1)) $$ = \frac{1}{2} L 2 \pi(5 n-3 n)=2 \pi n L$$
Q14. A planet spins about its axis with a period T. If its mass decreases by (2%) and radius increases by (3%), then its period will
1) increase by 1%
2) decrease by 1%
3) increase by 4%
4) decrease by 8%
Hint: (\tau=0) (L=I \omega=) constant (\frac{2}{5} MR^2 \frac{2 \pi}{T}=C) (\therefore \frac{MR^2}{T}=C) (\therefore T \propto MR^2) (M \downarrow es by 2% and R \uparrow es by 3%) (\therefore T changes by [-2%+(2 \times 3%)]=4%)
Q15. A disc of moment of inertia I rotates about a horizontal, axis with frequency n . A blob of wax of mass m falls vertically as shown in the figure, with constant velocity v . It sticks to the rim of disc. The frequency of rotation of the disc now is
1) (\frac{I R \omega-m v R}{I+m R^2})
2) (\frac{\omega+\frac{m v R}{I}}{I+\frac{m R^2}{I}})
3) (\frac{I \omega-m v R}{I+m R^2})
4) (\frac{I \omega}{I+m R^2})
Hint: (\tau=0) (on system) (L_i=L_f) (L_d+L_m=L_d'+L_m') (I \omega-mvR=I \omega'+mR^2 \omega') (\therefore \omega'=\frac{I \omega-mvR^2}{I+mR^2})
Q16. A ballet dancer spins with kinetic energy of 14 U. If her M.I. about same axis decreases by (30%), her kinetic energy would be
1) (\frac{10 U}{7})
2) 21 U
3) (\frac{140}{3} U)
4) 20 U
Hint: $$U=\frac{1}{2} \frac{L^2}{I} \Rightarrow U \propto \frac{1}{I} \\ \frac{U_2}{U_1}=\frac{I_1}{I_2} \\ = \frac{I_1}{\frac{70}{100} I_1}=\frac{100}{70}$$
(\therefore U_2=\frac{10}{7} U_1) $$= \frac{10}{7}(14 U)=20 U$$
Q17. The kinetic energy of a body, spinning in a friction free environment, changes by 2E when its angular speed changes from 30 rpm to 50 rpm . The change in its KE when angular speed changes from 50 rpm to 60 rpm is
Hint: (L=) constant (E=\frac{1}{2} L \omega) (\Delta E=\frac{1}{2} L \Delta \omega) (\frac{\Delta E_2}{\Delta E_1}=\frac{(\Delta \omega)_2}{(\Delta \omega)_1}) (\frac{\Delta E_2}{2 E}=\frac{(60-50) rpm}{(50-30) rpm}) (\Delta E_2=2 E \frac{10}{20}=E)
Q18. A toy helicopter is placed on a frictionless ice surface. When its rotor blades, start rotating in anticlockwise direction, with large speed, the body of the helicopter will
1) remain at rest.
2) rotate in anticlockwise.
3) rotate in clockwise direction.
4) move forward along ice surface.
Hint: (L=0) (L_{body}+L_{rotor}=L_{body}'+L_{rotor}') (0+0=L_{body}'+L_{rotor}') (\therefore L_{body}'=-L_{rotor}') ∴ The body starts rotating in anti-clockwise direction.
Q19. A particle of mass (m) is moving in a plane along a circle of radius (r). Its angular momentum about the axis of rotation is L . What is the centripetal force acting on the particle?
1) (\frac{L^2}{mr})
2) (\frac{L^2 m}{r})
3) (\frac{L^2}{m r^3})
4) (\frac{L^2}{mr^2})
Hint: (L=mvr) (v=\frac{L}{mr}) (F=\frac{mv^2}{r}=\frac{m}{r}[\frac{L}{mr}]^2=\frac{L^2}{mr^3})
Q20. If (L) and (E_R) represent the angular momentum and the rotational kinetic energy respectively of a body, then which of the following correctly represents the relation between L and (\sqrt{E_R}) ?
Hint: (E=\frac{L^2}{2 I} \Rightarrow \sqrt{E} \propto L)
Q21. When the M.I of a body increases by (50%), its angular momentum increases by (30%). The kinetic energy of the body will
1) increase by 12%.
2) decreases by 16%.
3) decreases by 26%.
4) increase by 26%.
Hint: (E=\frac{1}{2} \frac{L^2}{I} \Rightarrow E \propto \frac{L^2}{I}) (E \rightarrow \frac{1.69}{1.5}=1.126) (E \uparrow) by 12%
Q22. The kinetic energy of a rotating body is kept constant and the graph of angular momentum against frequency of rotation is plotted. The graph is
1) a straight line with positive slope.
2) a straight line with negative slope.
3) a parabola through origin symmetric about frequency axis.
4) a rectangular hyperbola.
Hint: (E=\frac{1}{2} L \omega=2 \pi Ln) (L \propto \frac{1}{n})
Q23. A disc X , of moment of inertia, (2 kg-m^2), rotating with a speed of 180 rpm , is coupled to another disc, Y , of moment of inertia, (1 kg-m^2), and at rest. They attain a common angular speed after 1 ms . The angular velocity of the system after coupling is
1) (2 rad/s)
2) (2 \pi rad/s)
3) (4 \pi rad/s)
4) (4 rad/s)
Hint: (I_1 \omega_1+I_2 \omega_2=(I_1+I_2) \omega) $$\omega =\frac{I_{1} \omega_{1}+I_{2} \omega_{2}}{I_{1}+I_{2}} \\ =\frac{2(180)+1(0)}{2+1} \\ =\frac{360}{3}=120 rpm \\ \omega =2 \pi\left(\frac{120}{60}\right)=4 \pi rad/s$$
Q24. A disc X , of moment of inertia, (2 kg-m^2), rotating with a speed of 180 rpm , is coupled to another disc, Y , of moment of inertia, (1 kg-m^2), and at rest. They attain a common angular speed after 1 ms , the torque acting on disc X is
1) (0 N-m)
2) (2 \pi \times 10^3 N-m)
3) (-4 \pi N-m)
4) (-4 \pi \times 10^3 N-m)
Hint: (\tau=I \alpha=I \frac{(\omega_2-\omega_1)}{t}=\frac{2[4 \pi-2 \pi(\frac{180}{60})]}{1 \times 10^{-3}}) $$ = \frac{2(4 \pi-6 \pi)}{10^{-3}}=-4 \pi \times 10^3 N-m$$
Q25. A disc X , of moment of inertia, (2 kg-m^2), rotating with a speed of 180 rpm , is coupled to another disc, Y , of moment of inertia, (1 kg-m^2), and at rest. They attain a common angular speed after 1 ms . the torque acting on disc Y is
1) (0 N-m)
2) (4 \pi \times 10^3 N-m)
3) (4 \pi N-m)
4) (-2 \pi \times 10^3 N-m)
Hint: (\tau_Y=-\tau_x=-(-4 \pi \times 10^3)=4 \pi \times 10^3 N-m)
