**Object as point particles**

Object should be treated as point particles, i.e. dimensionless (as small as a point on a paper), unless otherwise stated.

**Forces**

- Weight (W): W = mg, always directed downward; depends on location (e.g. Moon).
- Tension (T)*: due to stretched strings, depends on the force exerted on the string.
- Elastic (spring): F = kx (Hooke’s law), where k is the spring constant (in Nm^-1).
- Normal reaction force (N or R)*: perpendicular to the surface of the body exerting the force.
- Drag forces*: air resistance, fluid resistance – against motion.

* Result of electromagnetic interactions between molecules.

Drag forces

- Air resistance: normally proportional to the speed.
- Friction: caused by asperities in the surfaces; not affected by area or speed.
- Dynamic (when moving): Fd = μdR, where μd is the coefficient of dynamic friction (a dimensionless scalar value).
- Static (when not moving): Fs ≤ μsR , where μs is the coefficient of static friction, given that μs > μd and Fs is equal to the “pull”, unless the pull is greater than μsR, in which case the object moves.

**Newton’s laws of motion**

- First law (Principle of inertia): “An object continues to remain stationary or to move at a constant velocity unless an external force acts on it”
- Consequence e.g.: Person in a car accelerating feels “thrown backwards”, because the body would naturally maintain its state of motion.

- Second law: “F = ma” (simple form), where m is the body’s mass, a its acceleration (normally measured in ms^-2) and F the force acting on it (measured in N – newtons).
- Force and acceleration have the same direction, since they are both vectors.
- Net force = Resultant force = The sum of all forces =∑F
- When the speed is constant the resultant force is equal to zero.

- Third law: “Every action has an equal and opposite reaction. The action-reaction pair must be of the same type”. Hence, Fab = -Fba (Negative sign when against motion!)
- E.g. Gravitational force: “Pull of Earth on man” and “Pull of man on Earth”.

**Inclined plane:**

- Weight is decomposed into a component horizontal to the plane and a component vertical to the plane.
- Vertical: N= mg cos θ
- Horizontal: ∑F = mg sin θ – Fd

Free-body diagrams

- Illustration of all forces acting only on a body as vectors (Remember how to represent vectors).
- All forces must be clearly labeled (e.g. Weight force/mg or Normal reaction force/R)
- All forces must start at the center of the body.

Translational equilibrium

- The body must be at rest or constant velocity, i.e. net force = 0 (circular motion not!)

- Using tension: horizontal and vertical equilibrium.
- T1 sin θ1 = T2 sin θ2
- T = T1 cos θ1 + T2 cos θ2

Elevator issue

The reaction force is what a weighing scale measures. This is called the apparent weight.

### LAWS OF MOTION

### ARISTOTLE’S FALLACY

### LINEAR MOMENTUM (P)

Basic concepts

Linear momentum (p): mass x velocity – “quantity of motion”.

Impulse (I): change in momentum.

Derivation from Newton’s Second law (assuming constant mass):

∑F = ma = m∆v/∆t = ∆p/∆t

Impulse = Area under force-time graph.

Units: kg m s^-1 or Ns

Principle conservation of linear momentum

“Momentum is always constant, if the net force on the system is zero (closed system)”

Kinetic energy may or may not be conserved in a collision.

Collisions type

Elastic: Kinetic energy is totally conserved.

Inelastic: Kinetic energy is not conserved.

Totally inelastic (or plastic): Maximum kinetic energy lost – Bodies stick together.

Explosive: m

_{1}v_{1}= -m_{2}v_{2}

**elastic and equal masses**

**elastic and unequal masses**

**totally inelastic**

“Real life cases”

Recoil of a gun: Explosive “collision”. Initial momentum = final momentum =a 0.

Gun – higher mass, less speed; Bullet – less mass, higher speed.

Water hoses: A = cross-sectional area; l = cylinder’s length; p = H20 density.

Mass of water loss per second: pAl.

Rockets: Explosive “collision” – Mass thrown in one direction, rocket travels in the other

As total mass decreases, rate of increase of speed (acceleration) increases.

Airbags: Increases person’s head impact time – rate of transfer of momentum decreases (impulse remains the same) – Average force reduces.

### NEWTON’S LAWS OF MOTION

#### FIRST LAW

#### SECOND LAW

#### THIRD LAW

### EQUILIBRIUM OF A PARTICLE

#### STABLE EQUILIBRIUM

#### UNSTABLE EQUILIBRIUM

#### NEUTRAL EQUILIBRIUM

### COMMON FORCES IN MECHANICS

#### WEIGHT

#### TENSION

#### NORMAL FORCE

#### SPRING FORCE

#### FRICTIONAL FORCE

#### PSEUDO FORCE

### CONSTRAINT MOTION

### FRAME OF REFERENCE

- Inertial frame of reference: These are frames of reference in which Newton’s laws hold good. These frames are at rest with each other or which are moving with uniform speed with respect to each other. All reference frames present on surface of Earth are supposed to be inertial frame of reference.
- Non – inertial frame of reference: Newton’s law do not hold good in non-inertial reference frame.
- All accelerated and rotatory reference frames are non – inertial frame of reference. Earth is a non-inertial frame.

### FREE BODY DIAGRAM (FBD)

- If M2 > M1 and they move with acceleration a

- If the pulley begins to move with acceleration f, downwards

- When the mass M1 moves upwards with acceleration a.

- When the mass M1 moves downwards with acceleration a.

- If (M2/M1 = sinθ) then the system does not accelerate.
- Changing position of masses, does not affect the tension. Also, the acceleration of the system remains unchanged.
- If M1 = M2 = M (say), then

- A person of mass M climbs up a rope with acceleration a. The tension in the rope will be M(g+a).

- If the person climbs down along the rope with acceleration a, the tension in the rope will be M(g–a).

- When the person climbs up or down with uniform speed, tension in the string will be Mg.

- Acceleration down the plane is g sin θ.
- Its velocity at the bottom of the inclined plane will be
- Time taken to reach the bottom will be

- If angles of inclination are θ1 and θ2 for two inclined planes

- When lift is accelerated upward

- When lift is accelerated downward

- When lift is at rest or moving with constant velocity

N –mg = 0 or N = mg

- When a man jumps with load on his head, the apparent weight of the load and the man is zero.
- If a person sitting in a train moving with uniform velocity throws a coin vertically up, then coin will fall back in his hand.
- If the train is uniformly accelerated, the coin will fall behind him.
- If the train is retarded uniformly, then the coin will fall in front of him.

### LAW OF CONSERVATION OF LINEAR MOMENTUM

(i.e., p1, p2………), but their total momentum remains constant.

#### GUN FIRING A BULLET

### IMPULSE

#### ROCKET PROPULSION (A CASE OF SYSTEM OF VARIABLE MASS)

### FRICTION

- The force of static friction between any two surfaces in contact is opposite to and given by and (when the body just moves in the right direction).

- The force of kinetic friction is opposite to the direction of motion and is given by fk = μkN

where μk is coefficient of kinetic friction. - The value of μk and μs depends on the nature of surfaces and μk is always less then μs.

#### FRICTION ON AN INCLINED PLANE

#### SOME FACTS ABOUT FRICTION

- The force of kinetic friction is less than the force of static friction and the force of rolling friction is less than force of kinetic friction i.e.,

- Frictional force does not oppose the motion in all cases, in fact in some cases the body moves due to it.

#### LAWS OF LIMITING FRICTION

- The force of friction is independent of area of surfaces in contact and relative velocity between them (if it is not too high).
- The force of friction depends on the nature of material of surfaces in contact (i.e., force of adhesion). μ depends upon nature of the surface. It is independent of the normal reaction.
- The force of friction is directly proportional to normal reaction i.e., F ∝ N or F = mn.

#### ROLLING FRICTION

### CONSERVATIVE AND NON-CONSERVATIVE FORCES

### CASES OF CIRCULAR MOTIONS

#### MOTION IN A VERTICAL CIRCLE

- The body will complete the vertical circle if its velocity at lowest point is equal to or greater than
- The body will oscillate about the lowest point if its velocity at lowest point is less then . This will happen when the velocity at the halfway mark, i.e.
- The string become slack and fails to describe the circle when its velocity at lowest point lies between

#### CASE OF CYCLIST

#### CASE OF CAR ON A LEVELLED ROAD

#### CASE OF BANKING OF ROAD (FRICTIONLESS)

#### CASE OF BANKING OF ROAD (WITH FRICTION)

### CONICAL PENDULUM

- It is impossible for the string to be horizontal.

- The tension is always greater than mg.

- The tension can be calculated without knowing the inclination of the string since, from eqn. (2) and (3) ⇒
- The vertical depth h of P below A is independent of the length of the string since from eqn. (1) and (4) but