System of Particles and Rotational Motion

Rigid Body

Ideally, a rigid body is one with a perfectly definite and unchanging shape.

Translatory Motion

linear motion

In _________ motion, every particle of the body has the same velocity.

translatory motion

Rolling Motion

motion involving translatory and rotating motion

In ________________, every particle of the body has different velocity.

rolling motion

Rotational Motion

motion of a body around a fixed axis

examples for rotational motion

celling fan, potter's wheel

Precession

Precession is a type of rotational motion around a pivoted axis where the axis is free to move.

examples for precision

top and pedal fan

Centre of Mass

It is an imaginary point in a body where the entire mass of the body is said to be concentrated.

R꜀ₒₘ =

m₁r₁ + m₂r₂ +...+ mₙrₙ / m₁ + m₂ +...+ mₙ

In a regular body, the centre of mass is said to be the

centre of gravity

R꜀ₒₘ for a two particle system with equal mass =

(r₁ + r₂) / 2

R꜀ₒₘ for a body in an XY plane

X꜀ₒₘ = m₁x₁ + m₂x₂ +...+ mₙxₙ / m₁ + m₂ +...+ mₙ
= Σᵢⁿ₌₁ mᵢxᵢ/M
Y꜀ₒₘ = m₁y₁ + m₂y₂ +...+ mₙyₙ / m₁ + m₂ +...+ mₙ
= Σᵢⁿ₌₁ mᵢyᵢ/M
R꜀ₒₘ = (X꜀ₒₘ, Y꜀ₒₘ)

COM moves in the direction of the

external force

Velocity of COM

velocity of each particle/total mass

Acceleration of COM

acceleration of each particle/total mass

Force of COM

total force by each particle

Momentum of COM

total momentum by each particle

Vector Product of A and B

|A| ⋅ |B| ⋅ sinθ ⋅ n^

Cross product always gives _______ as result

vector

A x B (commutative)

is not equal to B x A

A x (B + C)

= A x B + A x C

A x A

= 0

î x î = j x j = k x k

= 0

î x j

= k

j x k

= î

î x k

= j

A x B

- ( B x A )

Torque

is the rotational effect of force. It is the product of force and the perpendicular displacement from its axis of rotation.

Torque is also called

moment of force or couple

Unit of Torque

Nm

Dimension of Torque

[M L² T⁻²]

τ =

r x F ( r ⋅ F ⋅ sinθ )

ω =

dθ/dt (v/r)

α =

dω/dt

Angular momentum

product of linear momentum and the perpendicular distance from the axis of rotation

l =

r x p

unit of angular momentum

kgm^2/s

Dimension of angular momentum

[ML²T⁻¹]

τ = dl / dt

d ( r x p ) / dt
= p dr/dt + r dp/dt
= p x v + r x F
= m x v x v + r x F
= 0 + r x F
= r x F = τ

Law of Conservation of Angular Momentum

When the τₑₓₜ applied on a body is equal to 0, the angular momentum remains constant.
Example: When a diver dives is about to throw himself into the water, he folds himself by hugging his calf muscles and jumps (to ↓ I and ↑ ω). Before reaching the water, he unfolds himself to ↑ I and ↓ ω, to easily land of water.

Equilibrium of rigid bodies in rotational motion

τₑₓₜ = 0;
l is a constant;
ω is a constant;
α = 0

Equilibrium of rigid bodies in translatory motion

Fₑₓₜ = 0;
p = constant;
v = constant;
a = 0

When is a rigid body in mechanical equilibrium?

A rigid body is said to be in mechanical equilibrium when both its angular momentum and linear momentum are constants.

Couple

A couple is a pair of equal and opposite forces acting on a body, but not along the same line.

________ causes rotational motion without translatory motion

A couple

Examples of couples

seesaw, magnetic compass

Principle of momentum

Consider a lever at some origin (Fulcrum) in mechanical equilibrium. Let f₁ and f₂ be the forces acting on A and B. Let R be the normal reaction at the fulcrum.
R - f₁ - f₂ = 0 (mechanical equilibrium)
R = f₁ + f₂

for rotational equilibrium, we take the moments about the fulcrum as zero.
τ₁ = -d₁ (- f₁) = d₁ f₁
τ₂ = d₂ (- f₂) = - d₂ f₂
τₜₒₜₐₗ = τ₁ + τ₂ = 0
= d₁ f₁ - d₂ f₂ = 0
d₁ f₁ = d₂ f₂ (mechanical equilibrium)
d₁/d₂ = f₂/f₁

Moment of Inertia

It gives the measure of inertia experienced during rotational motion. It is the product of the mass and square of the distance of the body from the axis aka radius of gyration (k)

I =

mk²

Radius of Gyration

distance from the centre of mass to the rotational axis

Unit of Moment of Inertia

kgm²

Dimesion of Moment of Inertia

[ML²]

Dimension of Radius of Gyration

[M⁰ L¹ T⁰]

Factors on which I depends

- mass of the body
- size and shape
- position and orientation of the rotational axis
- distribution of mass about the rotational axis

What is a flywheel? Where is it used?

A flywheel is a heavy disc attached to the end of a rotating shaft of the engine in an car. It smooths out the engine's power pulses and stores energy through rotational momentum.

Kinematic equations in rotational motion

ωₜ = ω₀ + αt
θ = ω₀t + ½ αt²
ωₜ² - ω₀² = 2αθ

ω in terms of ν

2πν

Dynamic equation for rotational motion

τ = Iα
τ = dl/dt
l = Iω
Work Done = τθ
Power = τω
KE = ½ I ω²

Kinetic energy of rolling motion (KEᵣₒₗₗ)

KEᵣₒₗₗ = KEᵣₒₜ + KEₜᵣ
= ½ I ω² + ½ mv²
= ½ mk² (v²/r²) + ½ mv²
= ½ mv² (k²/r² + 1)
= KEₜᵣ (k²/r² + 1)

rotational analog of force

torque

rotational analog of linear momentum

angular momentum

Rotational analog of mass is

Moment of Inertia