7- Torque

What is the formula for the angular momentum L⃗ of a point particle?

Its the cross-product between its radius vector r⃗ and its momentum p⃗

What is the formula for the rate of change of angular momentum of a point particle?

F⃗ = the force acting on the particle.

This product is called torque Γ⃗

What is the relationship between torque and the rate of change of angular momentum for a point particle?

The rate of change of angular momentum of a point particle is equal to the torque acting on the particle

What is the total force acting on each particle in a system of particles?

The sum of the external force and the internal forces from other particles

What is the formula for the sum of the internal torques for two interacting particles?

What are central forces, and how do they affect the sum of internal torques?

• They forces that act along the line joining two particles.

• When forces are central, the cross product (r⃗_j−r⃗_k)×F⃗_jk​ becomes zero, and thus the internal torques cancel each other out.

What is the final form of the equation for the rate of change of angular momentum in a system of particles or rigid body?

What is the consequence of having zero net external torque acting on a system or rigid body?

If the net external torque is zero, the angular momentum of the system or rigid body is conserved.

What is the definition of the centre of gravity of a body?

The centre of gravity is the point about which the resultant torque due to the gravitational force is zero.

What is the condition for the net gravitational torque when the origin of the reference frame is at the centre of mass?

• The net gravitational torque is zero when the reference frame is at the centre of mass

• This is because the sum of the position vectors of all particles weighted by their masses is zero.

What is formula for the net torque about a point C shifted from the centre of mass by a vector R?

What is the relationship between angular momentum and angular velocity for a fixed axis rotation?

L_z​ = the angular momentum

I =the moment of inertia

ω= the angular velocity.

How is the rate of change of angular momentum related to torque in fixed axis rotation?

What is the definition of angular acceleration?

Angular acceleration β is the rate of change of angular velocity

What is the equation relating moment of inertia, angular acceleration, and external torque in fixed axis rotation?

I= the moment of inertia

β=the angular acceleration

Γ_z(e) =the external torque along the axis.

What condition must be met for rotational equilibrium in fixed axis rotation?

The net external torque must have zero projection onto the axis of rotation, meaning no net torque acting on the body.

What happens if the net external torque is zero in a fixed axis rotation?

A body with a fixed axis will remain in a state of uniform rotation (fixed angular velocity) or a state of rest (zero angular velocity).

How is the total kinetic energy of a rotating body expressed?

r⊥_,j​ = the perpendicular distance from the particle to the axis of rotation

ω= is the angular velocity.

What is the formula for the moment of inertia in terms of the distance from the axis of rotation?

r⊥_,j​ =the distance of each particle from the axis of rotation.

What is the formula for the total kinetic energy of a rigid body rotating around a fixed axis?

I = the moment of inertia

ω=the angular velocity.

What is the formula for the work done by the net external torque is?

Its equal to the change in its kinetic energy

ω1​ and ω0are the final and initial angular velocities.