Short Notes: Laws of Motion

Table of Contents
1. 4.1 Newton's Laws
2. 4.2 Friction
3. 4.3 Motion on an Inclined Plane
4. 4.4 Connected Bodies
5. 4.5 Impulse and Momentum
6. Applications and problem tips
View more

4.1 Newton's Laws

Mechanics begins with the concept of force, a vector quantity that tends to produce acceleration of a body. Mass is the measure of inertia, the property of a body to resist change in its state of motion. The three fundamental statements that relate force, mass and motion are known as Newton's laws of motion. These laws form the basis for solving most problems on dynamics.

First Law (Law of Inertia)

The first law states that if the resultant external force on a body is zero, its velocity remains constant. This explains why a moving object continues to move in a straight line with constant speed unless a net force acts on it. Inertia is the resistance to change in motion; mass quantifies inertia.

Second Law (Relation between force and motion)

The second law provides a quantitative relation between net external force and motion. If a constant net force acts on a body of mass m, the acceleration a produced is given by F = ma. More generally, the rate of change of momentum p = mv equals the net external force: F = dp/dt. This law defines the direction of acceleration as the direction of the net force.

Third Law (Action-Reaction)

The third law states that forces always occur in pairs. If body A exerts a force on body B, then body B exerts an equal and opposite force on body A. These two forces act on different bodies and therefore do not cancel each other.

4.2 Friction

Friction is the resistive force that acts parallel to the contacting surfaces and opposes relative motion or the tendency to move. Friction depends on the nature of surfaces and the normal reaction between them, not on the contact area (for typical dry surfaces).

• μ_s and μ_k are the coefficients of static and kinetic friction respectively.
• Limiting friction is the maximum static friction just before sliding starts; beyond this, motion begins and kinetic friction applies.
• Angle of repose θ is the steepest angle at which a body rests on an inclined plane without sliding. It satisfies tan θ = μ_s.

4.3 Motion on an Inclined Plane

Resolve the weight mg of a block on a plane inclined at angle θ into components parallel and perpendicular to the plane.

• Component parallel to plane: mg sin θ
• Component perpendicular to plane: mg cos θ
• Normal reaction: N = mg cos θ
• With kinetic friction coefficient μ, acceleration down the plane: a = g(sin θ - μ cos θ)
• Acceleration up the plane when block is being pulled up with friction resisting motion: a = -g(sin θ + μ cos θ)

Equilibrium and limiting conditions

• If no friction is present, acceleration down the plane is a = g sin θ.
• If static friction prevents motion, the block remains at rest when mg sin θ ≤ f_s,max = μ_s mg cos θ, i.e. when tan θ ≤ μ_s.

Derivation: acceleration with kinetic friction (concise)

The following steps show how the expression for acceleration down the plane with kinetic friction is obtained.

Resolve forces along the plane.

The net force along the plane is the component of weight down the plane minus kinetic friction.

Net force = mg sin θ - f_k

f_k = μ N = μ mg cos θ

Therefore net force = mg sin θ - μ mg cos θ

Using F = ma gives:

a = g(sin θ - μ cos θ)

4.4 Connected Bodies

Systems of connected bodies commonly appear in problems involving strings and pulleys. For ideal strings (massless, inextensible) and massless frictionless pulleys, acceleration and tensions can be found using Newton's second law applied to each mass.

• For two masses m₁ and m₂ connected over a pulley (m₁ > m₂): a = (m₁ - m₂) g / (m₁ + m₂)
• Tension in the string for the two-mass system: T = 2 m₁ m₂ g / (m₁ + m₂)
• For two masses m₁ and m₂ on a frictionless horizontal surface connected by a string pulled by a force F: a = F / (m₁ + m₂)

Derivation: two masses over a pulley

The derivation below shows the standard method to obtain acceleration and tension for two masses connected over an ideal pulley.

Take m₁ to move down and m₂ up, so acceleration of both is a (same magnitude).

For m₁: m₁ g - T = m₁ a

For m₂: T - m₂ g = m₂ a

Adding the two equations gives:

(m₁ - m₂) g = (m₁ + m₂) a

Therefore:

a = (m₁ - m₂) g / (m₁ + m₂)

Substitute a back into one of the earlier equations to obtain T. Using m₁ g - T = m₁ a:

T = m₁ g - m₁ a

T = m₁ g - m₁ (m₁ - m₂) g / (m₁ + m₂)

Simplifying yields:

T = 2 m₁ m₂ g / (m₁ + m₂)

4.5 Impulse and Momentum

Momentum of a particle is defined as the product of its mass and velocity. Momentum is a vector quantity and is denoted by p.

Impulse measures the effect of a force acting over a short time interval and equals the change in momentum. In the absence of external forces, the total momentum of a system remains constant. This principle is used to analyse collisions and explosions.

Types of collisions (brief)

• Elastic collision: kinetic energy and momentum are conserved.
• Inelastic collision: momentum is conserved but kinetic energy is not; objects may stick together in a completely inelastic collision.

Example: impulse when velocity changes

Consider a particle of mass m whose velocity changes from u to v due to a force acting for time Δt. The impulse delivered is the change in momentum.

Impulse, J = m(v - u)

Also J = F Δt

Applications and problem tips

• Always start by drawing clear free-body diagrams showing all forces (weight, normal reaction, friction, tension, applied forces).
• Choose convenient coordinate axes (often along and perpendicular to an inclined plane) and resolve forces into components.
• For connected bodies use consistent sign conventions for acceleration and write Newton's second law for each mass before eliminating internal forces (tensions).
• Use conservation of momentum for collision problems only when external impulses are negligible during the short collision time.
• Remember that coefficients of friction depend on the pair of surfaces in contact and are determined experimentally; μ_k is generally smaller than μ_s.

Summary: Newton's three laws provide the framework for analysing forces and motion. Friction opposes relative motion and affects acceleration on surfaces and inclined planes. Connected-body problems require simultaneous application of Newton's second law to each part. Impulse and momentum are central for understanding collisions and sudden changes in motion.