Question

I'm trying to make a study sheet, what are the formula for

Kinematics in one dimension,
Vectors, kinematics in two dimensions,
Dynamics, Newton’s laws of motion.
Applications of Newton’s laws, gravity,
Work and energy,
Conservation of energy,
Linear momentum and collisions,
Rotational motion,
Static equilibrium, elasticity, fracture,
Fluids
Heat and temperature; kinetic theory of gases,
Thermodynamics

Answer 1

1.Kinematics in one dimension:

The kinematic equations for straight line motion with constant acceleration are:

1..2..3.. 4..

2. Vectors:

Addition of vectors:

;

Resolution of Vectors:

.

The scalar product of 2 vectors: where a is the magnitude of ,b is the magnitude of and is the angle between the 2 vectors.

3. Kinematics in 2 dimensions:

Motion in a plane with constant acceleration

.  .

Uniform circular motion

The centripetal acceleration of a body moving in a circle of radius r with constant speed v is .

Relative velocity and acceleration

where is the velocity of the particle relative to the S-frame, is the velocity of the particle relative to the S'-frame and is the velocity of the S'-frame relative to the S-frame.

4.Dynamics- Newton's laws of motion.

The fundamental equation of classical mechanics:

.

Weight and mass:.

Frictional force:

The ratio of the magnitude of the maximum force of static friction to the magnitude of the normal force is the coefficient of static friction i.e..

For a body undergoing uniform circular motion, the centripetal force .

5. Work and energy.

Let a constant force make an angle with the x-axis and act on a particle whose displacement along the x-axis is .

Then the work done by the force .

Kinetic energy and the Work-Energy theorem:

The kinetic energy of a body where m is the mass of the body and v is the velocity of the body.

The work-energy theorem states that the work done by the resultant force acting on a particle is equal to the change in the kinetic energy of the particle. i.e .

Power:

Power is the time rate at which work is done. i.e.

6.Conservation of Energy.

For conservative forces, the total mechanical energy is constant i.e..

The relation between force and potential energy for one-dimensional motion is:

.

The conservation of energy principle states that the total energy-kinetic plus potential plus heat plus all other forms-does not change. Energy may be transformed from one kind to another, but it cannot be created or destroyed; the total energy is constant.

7. Linear momentum and collisions.

Center of Mass:

For a large number of particles distributed in space .

The above 3 scalar equations can be replaced by a single vector equation:

.

The center of mass of a system of particles depends only on the masses of the particles and the positions of the particles relative to one another.

Linear momentum of a particle:

The momentum of a single particle is a vector defined as the product of its mass m and its velocity i.e .

Also the rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the direction of that force i.e..

Linear momentum of a system of particles:
i.e the total momentum of a system of particles is equal to the product of the total mass of the system and the velocity of its center of mass.

Also Newton's second law for a system of particles iis where is the vector sum of all the external forces acting on the system.

The principle of the conservation of linear momentum states that when the resultant external force acting on a system is zero, the total vector momentum of the system remains constant.i.e.

Collisions:
The integral of a force over the time interval during which the force acts is called the impulse of the force. The change in momentum of a body acted on by an impulsive force is equal to the impulse i.e.

.

Conservation of momentum during collisions:

If there are no external forces the total momentum of a system is not changed by the collision.

When kinetic energy is conserved, the collision is said to be elastic, Otherwise the collision is inelastic.

8.Rotational Motion.

Rotational kinematics:

For a pure rotation of a rigid body about a fixed axis:
The instantaneous angular speed and the instantaneous angular acceleration is .

The kinematical equations describing this motion are:

1. 2. 3. 4..

Relation between linear and angular kinematics for a particle in circular motion:

;; and .

Rotational dynamics:

For a single particle:

The torque acting on a particle with respect to the origin O is defined as: .

The angular momentum of a particle with respect to the origin O is defined as:

l=r*p.

Also the time rate of change of the angular momentum of a particle is equal to the torque acting on it.

For a system of particles the time rate of change of the total angular momentum about the origin of an inertia reference frame is equal to the sum of the external torques acting on the system.

The expression for the kinetic energy of a rotating rigid body is K=1/2Iw² where I is the moment of inertia of the body.

Conservation of angular momentum:

9. Fluids.

Statics:

For a homogeneous liquid p₂-p₁=-dg(y₂-y₁) where p₁ is the pressure at elevation y₁ and p₂ is the pressure at elevation y₂.

Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.

Archimides's principle states that a body wholly or partly immersed in a fluid is buoyed up with a force equal to the weight of the fluid displaced by the body.