Level 1 — RecognitionRotational Mechanics

Rotational Mechanics

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Level 1: Recognition (MCQ + Matching + True/False with Justification)

Time Limit: 20 minutes
Total Marks: 30

Section A — Multiple Choice (1 mark each) [10 marks]

Choose the correct option.

Q1. The relation between linear velocity $v$ and angular velocity $\omega$ for a point at distance $r$ from the axis is: (a) $v = \omega / r$ (b) $v = r\omega$ (c) $v = r\omega^2$ (d) $v = r^2\omega$

Q2. The moment of inertia of a solid sphere of mass $M$ and radius $R$ about its diameter is: (a) $\frac{1}{2}MR^2$ (b) $\frac{2}{3}MR^2$ (c) $\frac{2}{5}MR^2$ (d) $MR^2$

Q3. A rigid body free in space has how many degrees of freedom? (a) 3 (b) 4 (c) 6 (d) 9

Q4. Torque is defined as: (a) $\vec{r}\cdot\vec{F}$ (b) $\vec{r}\times\vec{F}$ (c) $\vec{F}\times\vec{r}$ giving same sense as (b) (d) $\vec{F}/\vec{r}$

Q5. The rotational kinetic energy of a body is: (a) $\frac{1}{2}I\omega$ (b) $I\omega^2$ (c) $\frac{1}{2}I\omega^2$ (d) $\frac{1}{2}mv^2$

Q6. For rolling without slipping of a wheel of radius $R$ , the condition connecting centre velocity $v$ and $\omega$ is: (a) $v = \omega/R$ (b) $v = R\omega$ (c) $v = R^2\omega$ (d) $v = 2R\omega$

Q7. The parallel axis theorem is stated as: (a) $I = I_{CM} - Md^2$ (b) $I = I_{CM} + Md^2$ (c) $I = I_{CM} + Md$ (d) $I_z = I_x + I_y$

Q8. Angular momentum for a body rotating about a fixed axis is: (a) $L = I/\omega$ (b) $L = I\omega$ (c) $L = \frac{1}{2}I\omega^2$ (d) $L = r\times p$ only

Q9. Angular momentum is conserved when the net external torque is: (a) maximum (b) constant and non-zero (c) zero (d) equal to $I\alpha$

Q10. The perpendicular axis theorem $I_z = I_x + I_y$ applies only to: (a) any 3D body (b) planar (laminar) bodies (c) spheres (d) rods

Section B — Matching (1 mark each pair) [8 marks]

Match each body/axis in Column I with its moment of inertia in Column II.

Column I	Column II
P. Thin rod about centre	1. $\frac{2}{3}MR^2$
Q. Ring about central axis	2. $\frac{1}{2}MR^2$
R. Solid disk about central axis	3. $\frac{1}{12}ML^2$
S. Hollow sphere about diameter	4. $MR^2$

Q11. P → ?
Q12. Q → ?
Q13. R → ?
Q14. S → ?

Section C — True/False WITH Justification (2 marks each: 1 verdict + 1 reason) [12 marks]

Q15. "The centripetal acceleration of a point on a rotating body is $a_c = r\alpha$ ."

Q16. "For a rolling body descending an incline, a solid sphere reaches the bottom faster than a hollow cylinder (same mass, radius, height)."

Q17. "The torque equals the rate of change of angular momentum, $\vec{\tau} = d\vec{L}/dt$ ."

Q18. "A rigid body is in complete equilibrium if the net force on it is zero."

Q19. "In the parallel axis theorem, $d$ is the distance between the axis through the centre of mass and the parallel axis."

Q20. "The total kinetic energy of a rolling body is $\frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$ ."

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1. (b) $v = r\omega$ — tangential speed scales linearly with radius. (1)

Q2. (c) $\frac{2}{5}MR^2$ — standard result for solid sphere about diameter. (1)

Q3. (c) 6 — three translational + three rotational degrees of freedom for a free rigid body. (1)

Q4. (b) $\vec{\tau} = \vec{r}\times\vec{F}$ — cross product; magnitude $rF\sin\theta$ . (1)

Q5. (c) $\frac{1}{2}I\omega^2$ — rotational analogue of $\frac{1}{2}mv^2$ . (1)

Q6. (b) $v = R\omega$ — contact point instantaneously at rest. (1)

Q7. (b) $I = I_{CM} + Md^2$ . (1)

Q8. (b) $L = I\omega$ for fixed axis. (1)

Q9. (c) zero — no external torque ⇒ $L$ constant. (1)

Q10. (b) planar (laminar) bodies — theorem derived assuming mass lies in one plane ( $z=0$ ). (1)

Section B (1 mark each)

Q11. P → 3 ( $\frac{1}{12}ML^2$ , rod about centre) (1)
Q12. Q → 4 ( $MR^2$ , ring) (1)
Q13. R → 2 ( $\frac{1}{2}MR^2$ , disk) (1)
Q14. S → 1 ( $\frac{2}{3}MR^2$ , hollow sphere) (1)

Section C (1 verdict + 1 reason)

Q15. FALSE. (1) Centripetal (radial) acceleration is $a_c = r\omega^2 = v^2/r$ ; $r\alpha$ is the tangential acceleration $a_t$ . (1)

Q16. TRUE. (1) Acceleration on incline $a = \dfrac{g\sin\theta}{1 + I/mR^2}$ . Solid sphere $I/mR^2 = 2/5 = 0.4$ ; hollow cylinder $I/mR^2 = 1$ . Smaller factor ⇒ larger $a$ ⇒ sphere is faster. (1)

Q17. TRUE. (1) Newton's second law for rotation: net external torque equals the time rate of change of angular momentum. (1)

Q18. FALSE. (1) Full equilibrium requires both $\sum\vec{F}=0$ (translational) and $\sum\vec{\tau}=0$ (rotational); zero force alone allows a net couple to rotate the body. (1)

Q19. TRUE. (1) $d$ is the perpendicular distance between the CM axis and the chosen parallel axis; $I = I_{CM}+Md^2$ . (1)

Q20. TRUE. (1) Total KE = translational KE of CM ( $\frac{1}{2}mv^2$ ) + rotational KE about CM ( $\frac{1}{2}I\omega^2$ ). (1)

[
  {"claim":"Solid sphere I/mR^2 = 2/5",
   "code":"I=Rational(2,5); result = (I == Rational(2,5))"},
  {"claim":"Sphere acceleration exceeds hollow cylinder on incline",
   "code":"g,th=symbols('g theta',positive=True); a_sph=g*sin(th)/(1+Rational(2,5)); a_cyl=g*sin(th)/(1+1); result = simplify(a_sph - a_cyl) > 0 if False else bool((a_sph.subs({g:9.8,th:0.5}) - a_cyl.subs({g:9.8,th:0.5}))>0)"},
  {"claim":"Rod about centre = 1/12 M L^2",
   "code":"M,L=symbols('M L',positive=True); I=integrate((M/L)*x**2,(x,-L/2,L/2)); result = simplify(I - M*L**2/12)==0"},
  {"claim":"Parallel axis: disk about edge = 3/2 M R^2",
   "code":"M,R=symbols('M R',positive=True); Icm=M*R**2/2; I=Icm+M*R**2; result = simplify(I - Rational(3,2)*M*R**2)==0"}
]