Laws of Motion - Class 11 Physics

Laws of Motion

Overview: Dynamics deals with the causes of motion (Force). This chapter covers Newton's three laws, momentum, friction, and circular motion dynamics.

1. Concept of Force & Inertia

Force: An external effort in the form of push or pull that changes or tends to change the state of rest or of uniform motion of a body.

Inertia: The inherent property of a body to resist any change in its state of rest or uniform motion. It depends only on Mass.

2. Newton's Laws of Motion

First Law (Law of Inertia)

Everybody continues to be in its state of rest or of uniform motion unless compelled by some external force to act otherwise.

Second Law

The rate of change of momentum is directly proportional to the applied force definition.

F = dp/dt = ma

(If mass is constant)

Third Law

To every action, there is always an equal and opposite reaction. Forces always occur in pairs.

F_AB = -F_BA

3. Momentum & Impulse

Linear Momentum (p): Product of mass and velocity. p = mv.

Impulse (J): Large force acting for a short time. Change in momentum.

J = F × Δt = Δp

Law of Conservation of Momentum

If no external force acts on a system, the total linear momentum remains conserved.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

4. Friction

A contact force that opposes relative motion between surfaces.

Static Friction (f_s): Self-adjusting force up to a limit (Limiting Friction). f_s ≤ μ_sN.
Kinetic Friction (f_k): Opposes actual relative motion. f_k = μ_kN.
Rolling Friction: Much smaller than sliding friction.

Lubrication: Reduces friction by separating surfaces with a thin layer of oil/fluid.

5. Dynamics of Circular Motion

For a body to move in a circle, a net force must act towards the center.

F_c = mv² / R

Circular Turn on Level Road

Friction provides centripetal force.

v_max = √(μRg)

Banked Road

Road surface is inclined to reduce reliance on friction.

tanθ = (v² / Rg)

(For optimum speed where no friction is needed)

Numericals - Laws of Motion

Numericals

Newton's Second Law

Q1. A force of 50N acts on a mass of 10kg. Find the acceleration produced.

F = 50 N, m = 10 kg

F = ma → a = F/m

a = 50 / 10

a = 5 m/s²

Momentum Change

Q2. A cricket ball of mass 0.15 kg moving at 12 m/s is hit by a bat so that it turns back with speed 20 m/s. Find impulse.

m = 0.15 kg, u = 12 m/s, v = -20 m/s

Impulse I = Δp = m(v - u)

I = 0.15 × (-20 - 12)

I = 0.15 × (-32)

I = -4.8 Ns

Magnitude is 4.8 Ns.

Conservation of Momentum

Q3. A bullet of mass 0.04 kg moving at 90 m/s enters a heavy wooden block of 60 kg and is stopped after 60 cm. Find average resistive force.

Using Work-Energy theorem is easier here.

W = ΔK

-F × s = 0 - ½mu²

F × 0.6 = 0.5 × 0.04 × (90)²

0.6 F = 0.02 × 8100

0.6 F = 162

F = 162 / 0.6 = 270 N

Elevator Physics

Q4. A man of mass 70kg stands on a weighing scale in a lift accelerating upwards at 5 m/s². Reading? (g=10)

Effective gravity increases going up.

N = m(g + a)

N = 70(10 + 5)

N = 70 × 15

N = 1050 Newtons

Reading in kg = 105 kg.

Motion of Connected Bodies

Q5. Two masses 8kg and 12kg are connected by a string over a pulley. Find acceleration of the system.

m1 = 8kg, m2 = 12kg

a = (m2 - m1)g / (m1 + m2)

a = (12 - 8)10 / (12 + 8)

a = (4 × 10) / 20

a = 40 / 20

a = 2 m/s²

Recoil of Gun

Q6. A machine gun fires a bullet of mass 40g with a velocity 1200 m/s. The man holding it can apply max force 144N. Max bullets per sec?

Force = n × change in momentum per sec

F = n × mv

144 = n × 0.04 × 1200

144 = n × 48

n = 144 / 48

n = 3 bullets/sec

Friction on Incline

Q7. A block of mass 2kg slides down an incline of 30° with constant velocity. Find coefficient of kinetic friction.

Constant vy → Net force = 0

mg sinθ = f_k = μmg cosθ

μ = tanθ

μ = tan 30°

μ = 1/√3 ≈ 0.577

Banked Road

Q8. A circular track of radius 300m is banked at an angle of 15°. Optimum speed considering no friction? (tan 15 = 0.27)

v = √(Rg tanθ)

v = √(300 × 10 × 0.27)

v = √(3000 × 0.27)

v = √(810)

v ≈ 28.5 m/s

Tension in String

Q9. Three blocks of masses 2kg, 3kg, and 5kg are connected by strings and pulled by a 10N force on a smooth surface. Find value of acceleration.

a = Net Force / Total Mass

a = 10 / (2 + 3 + 5)

a = 10 / 10

a = 1 m/s²

Static Friction

Q10. A block of 10kg is on a rough floor (μs = 0.5). A force of 40N is applied. Will it move?

Max Static Friction f_lim = μN = μmg

f_lim = 0.5 × 10 × 10

f_lim = 50 N

Applied Force F = 40 N

Since F < f_lim, the block will NOT move.

Friction Force = 40 N (Self adjusting)

Formulas & Facts - Laws of Motion

Equations & Formulas

Concept	Formula
Momentum	p = mv
2nd Law	F = dp/dt = ma
Impulse	J = F × Δt = Δp
Limiting Friction	f_s = μ_sN
Kinetic Friction	f_k = μ_kN
Centripetal Force	F_c = mv²/R
Banked Road	tanθ = v²/Rg
Max Safe Speed	v = √[Rg(μ+tanθ)/(1-μtanθ)]
Connected Motion	a = Net Force / Total Mass

50 NEET Facts

Key points for Laws of Motion.

1. Inertia and Mass Mass is the measure of inertia. The heavier the body, the larger is its inertia.

2. No Force for Motion No force is required to move a body uniformly along a straight line. Force is only needed to change its motion.

3. Action-Reaction Action and reaction forces act on different bodies. Hence, they never cancel each other out.

4. Momentum Vector Momentum has the same direction as velocity. It is a vector quantity.

5. Impulse Graph Area under Force-Time graph gives the magnitude of Impulse (Change in Momentum).

6. Static Friction Static friction is a self-adjusting force. It changes its magnitude to balance the applied force up to a limit.

7. Kinetic vs Static Coefficient of kinetic friction is always less than coefficient of static friction (μk < μs).

8. Angle of Repose The angle of incline at which a block just begins to slide down is equal to the angle of friction. (tanα = μ).

9. Apparent Weight In a lift accelerating up, weight increases (mg+ma). Accelerating down, weight decreases (mg-ma). In free fall, weight is zero.

10. Horse Cart Problem The horse pushes the ground backward, ground pushes horse forward (Action-Reaction). Motion occurs because ground reaction > cart pull.

11. Sand on Belt If sand is dropped on a moving belt, force needed to keep velocity constant is F = v(dm/dt).

12. Rolling Friction Caused by deformation of surfaces. It is inversely proportional to radius of the wheel.

13. Rocket Propulsion Works on conservation of momentum. Thrust force F = u(dm/dt) where u is exhaust velocity.

14. Banking Angle Independent of the mass of the vehicle. Depends only on radius and speed.

15. Centrifugal Force A pseudo force experienced only in a rotating (non-inertial) frame. Directed outwards.

16. Tension in Rotary Motion Tension provides centripetal force for a stone tied to a string being whirled. T = mv²/r.

17. Vertical Circle Tension Tension is max at bottom (mg + mv²/r) and min at top (mv²/r - mg). Diff is 6mg.

18. Pulley Trick If a pulley moves with acceleration a_p, acceleration of blocks relative to ground changes. a_net = a_rel + a_p.

19. Spring Force F = -kx. Spring constant k depends on length. k ∝ 1/Length. Cutting spring in half doubles k.

20. Frame of Reference Newton's laws are only valid in Inertial Frames. For non-inertial, apply Pseudo Force (-ma) opposite to acceleration.

21. Friction Direction Friction opposes relative slipping. It can act in the direction of motion (e.g., for a person walking).

22. Catching a Ball Cricketer pulls hands back to increase time of impact, thereby reducing the impulsive force.

23. Explosion Internal forces cannot change net momentum of the system. Center of mass continues same path.

24. Lami's Theorem Application Useful for 3 concurrent forces in equilibrium. Avoids resolving components.

25. Bending of Cyclist Cyclist bends to shift Center of Gravity towards center of curvature to avoid toppling.

26. Cream Separator Works on principle of centrifuge. Heavier particles move out (large radius), lighter stay in.

27. Equilibrium Vector sum of all forces is zero. It does not mean body is at rest; it can be in uniform motion.

28. Contact Force Net contact force is vector sum of Normal Reaction and Friction.

29. Impulsive Force Gravity is not impulsive (magnitude is small compared to impact forces). Friction can be impulsive.

30. Pseudo Force Direction Always opposite to the direction of acceleration of the non-inertial frame.

31. Block on Block Friction acts between blocks to prevent relative sliding. They move together if F < F_max common.

32. Monkey Climbing If monkey accelerates up, Tension T = m(g+a). If down, T = m(g-a).

33. Breaking Strength Depends on material of rope, not on length. Breaking force = Stress × Area.

34. Skidding Occurs if centripetal force required > max static friction available.

35. Well of Death Vehicle is held by friction against gravity. Normal force provides centripetal force. μN = mg → μ(mv²/r) = mg.

36. Third Law Misconception Action and reaction do not cancel because they act on different objects. They produce different accelerations if masses differ.

37. Recoil Velocity V = -(m/M)v. Velocity of gun is opposite to bullet and much smaller due to large Mass M.

38. Polishing Polishing reduces friction initially but excessive polishing increases friction due to molecular bonding (sticking).

39. Streamlining Reduces fluid friction (drag) by giving bodies a shape that cuts through fluid easily (e.g. fish, airplanes).

40. Atwood Machine Acceleration a = [(m2-m1)/(m1+m2)]g. Tension T = [2m1m2/(m1+m2)]g.

41. Spring Balance Measures Tension/Force, not mass. Readings are in Newtons (or kg-wt).

42. Impulse-Momentum Eq Applicable for variable forces too integration F dt.

43. Normal Reaction Not always equal to mg. On incline N = mg cosθ. In lift N = m(g+a). N is a self adjusting force preventing penetration.

44. Friction Work Kinetic friction always does negative work (dissipates energy). Static friction does zero work in frame of surface.

45. Minimum Pull To pull a block, applying force at angle φ = tan^-1μ requires minimum force.

46. Walking on Ice Difficult because friction is low. We press hard to increase Normal and hence friction. Small steps help.

47. Newton's 1st vs 2nd 1st law defines force. 2nd law measures it.

48. Mass invariant Inertial mass (F/a) is same as Gravitational mass (W/g).

49. Area of Contact Friction is independent of area of contact (macroscopically), provided Normal Reaction is constant.

50. Fundamental Forces All contact forces (Friction, Normal, Tension) originate from Electromagnetic forces at atomic level.

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