CUET UG Physics Mechanics Laws of Motion Friction Circular Motion Class XI Revision

Must-Know

• Friction is a contact force that opposes relative motion between two surfaces; its maximum value is ( f_{\text{max}} = \mu_s N ), where ( \mu_s ) is the coefficient of static friction and ( N ) is normal reaction. Example: A 10 kg block on a horizontal surface with ( \mu_s = 0.5 ) requires at least 49 N force to start moving (since ( N = mg = 98\,N )).
• Kinetic friction is less than static friction: ( f_k = \mu_k N ), where ( \mu_k < \mu_s ); verified from NCERT experiments showing lower force needed to keep an object moving.
• Limiting friction is independent of the area of contact; doubling the surface area in contact does not change the maximum static friction if normal force remains constant.
• Rolling friction is much smaller than sliding friction; this is why wheels are used in transport—e.g., rolling friction coefficient for steel on steel is about 0.001 vs 0.75 for static sliding.
• Angle of repose (( \theta_r )) is the angle at which an object just begins to slide down an inclined plane: ( \tan \theta_r = \mu_s ); for ( \mu_s = 1 ), ( \theta_r = 45^\circ ).
• Centripetal force is not a new kind of force but the net force toward the center in circular motion; for a car turning on a level road, it is provided by static friction.
• Centripetal force formula: ( F_c = \frac{mv^2}{r} = m\omega^2 r ); e.g., a 1000 kg car moving at 10 m/s in a curve of radius 50 m needs ( 2000\,N ) centripetal force.
• On a banked road without friction, ideal speed is ( v = \sqrt{rg \tan \theta} ); for ( r = 100\,m ), ( \theta = 45^\circ ), ( v \approx 31.3\,m/s ).
• Banking reduces dependence on friction; at the ideal speed, no lateral friction acts on tires.
• In vertical circular motion, tension varies: at the lowest point, ( T = mg + \frac{mv^2}{r} ); at the highest point, ( T = \frac{mv^2}{r} - mg ).
• Minimum speed at the top of a vertical circle for a particle to complete the loop: ( v_{\text{min}} = \sqrt{gr} ); below this, the string goes slack.
• For a car on a convex bridge, apparent weight decreases: ( N = mg - \frac{mv^2}{r} ); at ( v = \sqrt{gr} ), ( N = 0 ) (feeling weightless).
• On a concave bridge, apparent weight increases: ( N = mg + \frac{mv^2}{r} ).
• Pseudo force acts in non-inertial frames; e.g., a person in an accelerating bus feels pushed backward, though no real force acts—this is due to inertia.
• The force required to move a block on a rough horizontal surface is minimum when the force is applied at an angle ( \theta = \tan^{-1}(\mu) ) to the horizontal.
• For a body sliding down a rough incline, acceleration is ( a = g(\sin\theta - \mu_k \cos\theta) ); if ( \theta = 30^\circ ), ( \mu_k = 0.2 ), then ( a \approx 3.2\,m/s^2 ).
• In uniform circular motion, velocity is tangential and acceleration is radially inward; direction changes continuously, so it is accelerated motion.
• Static friction adjusts up to a limit; if applied force is 5 N and ( f_{\text{max}} = 10\,N ), then static friction is exactly 5 N to prevent motion.
• Centripetal acceleration magnitude: ( a_c = \frac{v^2}{r} ); for Earth orbiting Sun (approx. circular), ( a_c \approx 0.006\,m/s^2 ).
• Impending motion means motion is about to start; at this point, static friction equals ( \mu_s N ).

Difficulty Level

Intermediate — combines conceptual understanding of forces with application in circular paths and inclined planes, requiring vector resolution and formula selection.

Common CUET Traps

• Trap: Assuming friction always opposes motion. Avoid: Friction opposes relative motion or impending motion; for a car accelerating forward, static friction on driving wheels acts forward.
• Trap: Using kinetic friction when motion hasn’t started. Avoid: Use static friction up to limiting value when object is at rest; only switch to kinetic when sliding occurs.
• Trap: Thinking centripetal force appears in free-body diagrams as a separate force. Avoid: Centripetal force is the net radial force; it must be the sum of real forces like tension, friction, or gravity.

Practice MCQs

1. A block of mass 5 kg is at rest on a horizontal surface. The coefficient of static friction is 0.4. What minimum horizontal force is needed to move it?
   A. 19.6 N
   B. 24.5 N
   C. 9.8 N
   D. 49 N
   Answer: A
   Explanation: ( f_{\text{max}} = \mu_s N = 0.4 \times 5 \times 9.8 = 19.6\,N ).
   Why others fail: D uses mass times g directly without multiplying by μ.

2. A car moves on a level circular track of radius 50 m. The coefficient of static friction between tires and road is 0.8. The maximum speed to avoid skidding is (g = 10 m/s²):
   A. 10 m/s
   B. 15 m/s
   C. 20 m/s
   D. 25 m/s
   Answer: C
   Explanation: ( v_{\text{max}} = \sqrt{\mu_s rg} = \sqrt{0.8 \times 50 \times 10} = \sqrt{400} = 20\,m/s ).
   Why others fail: A is too low, possibly from miscalculating square root or using wrong formula.

3. A cyclist is moving in a circular path of radius 30 m with speed 10 m/s. The angle of banking required for no friction is:
   A. ( \tan^{-1}(1/3) )
   B. ( \tan^{-1}(2/3) )
   C. ( \tan^{-1}(1/2) )
   D. ( \tan^{-1}(3/4) )
   Answer: A
   Explanation: ( \tan \theta = \frac{v^2}{rg} = \frac{100}{300} = \frac{1}{3} \Rightarrow \theta = \tan^{-1}(1/3) ).
   Why others fail: B comes from using v²/r without dividing by g.

4. A ball of mass 0.5 kg is tied to a string and whirled in a vertical circle of radius 1 m. What is the tension at the lowest point if speed is 6 m/s? (g = 10 m/s²)
   A. 18 N
   B. 23 N
   C. 28 N
   D. 33 N
   Answer: B
   Explanation: ( T = mg + \frac{mv^2}{r} = 5 + \frac{0.5 \times 36}{1} = 5 + 18 = 23\,N ).
   Why others fail: A forgets to add mg, using only centripetal term.

5. A block is placed on an inclined plane. The angle of inclination is gradually increased. It starts sliding when the angle is 37°. The coefficient of static friction is:
   A. 0.60
   B. 0.75
   C. 0.80
   D. 0.85
   Answer: B
   Explanation: ( \mu_s = \tan \theta = \tan 37^\circ \approx 0.75 ).
   Why others fail: A is tan 30°, a common confusion with standard angles.

Last‑Minute Revision

• ⚠️ Static friction ≤ μ_s N; equality holds only at impending motion.
• ⚠️ Kinetic friction = μ_k N; always less than limiting static friction.
• ⚠️ Angle of repose θ_r: tan θ_r = μ_s.
• ⚠️ Rolling friction < sliding friction < static friction (in magnitude).
• ⚠️ Centripetal force = mv²/r = mω²r — always directed toward center.
• ⚠️ Centripetal force is not a new force — it’s the net radial component.
• ⚠️ On banked road without friction: v = √(rg tan θ).
• ⚠️ Maximum speed on unbanked road: v_max = √(μ_s rg).
• ⚠️ In vertical circle, minimum speed at top = √(gr) for loop completion.
• ⚠️ Tension at lowest point = mg + mv²/r.
• ⚠️ Tension at highest point = mv²/r – mg.
• ⚠️ Apparent weight on convex bridge = m(g – v²/r).
• ⚠️ Apparent weight on concave bridge = m(g + v²/r).
• ⚠️ Pseudo force = –ma (in non-inertial frame).
• ⚠️ Optimal pulling angle: θ = tan⁻¹(μ) minimizes required force.
• ⚠️ Acceleration on rough incline: a = g(sinθ – μ_k cosθ).
• ⚠️ Friction adjusts to prevent motion until maximum value.
• ⚠️ Uniform circular motion has constant speed but variable velocity.
• ⚠️ Centripetal acceleration: a_c = v²/r ≈ 0.006 m/s² for Earth around Sun.
• ⚠️ Mnemonic: "Friction opposes relative motion" — not absolute motion.