One-Dimensional Kinematics


Two-Dimensional Kinematics

Newton’s Laws of Motion

Force is a push or a pull that causes objects to move.

SI derived unit of force is N (Newton)

1 N = 1


Work, Energy, and Power


Systems of Particles, Linear Momentum

System of Particles

Center of Mass: A single point where distribution of mass on all axis evens out; a single point whose translational motion is characteristic of the system as a whole.

Center of gravity is a point where distribution of gravitational force on all axis evens out; it will be the same point as the center of mass under uniform gravity.

Center of Mass of Multiple Particles

For 2 particles in a xy-plane,

Same can be done for the z-axis under xyz-plane problems.

For n particles:

To find the center of mass of a continuous body, you must integrate each mass element dm, as if they are infinite number of individual particles.

Momentum: Mass in motion

Momentum(kg*m/s) = Mass(kg) x Velocity(m/s)

Conservation of Momentum: Momentum is always conserved in any kind of collision.

Elastic Collision: A collision where both momentum and kinetic energy is conserved.

Inelastic Collision: A collision where momentum is conserved, but kinetic energy is not. A perfectly inelastic collision would be for the colliding objects to stick together, where maximum kinetic energy of a system is lost. Note that kinetic energy is not destroyed, as that would break the law of conservation of energy

Force is the change in momentum:

assuming ,

For  to stand true, in addition, mass must be assumed to stay constant.

Impulse: The change in momentum when force is acted upon an object, over an interval of time.

 or  

Impulse is most commonly seen in cars on impact;

A high force, sudden impact, is what usually kills. Because the momentum change acted upon cannot be changed, effort is put in to increase the time period to reduce the force.

For example, cars are made to crumple on impact, so that the force on impact can be distributed over a longer time.


Circular Motion and Rotation

What is Circular Motion?

Uniform Circular Motion:

Centripetal Acceleration and Centripetal Force:

Non-Uniform Circular Motion:

 

Horizontal Circular Motion:

Conical Pendulum:

 supports the weight.

Car on Horizontal Road:

Banked Frictionless Road & Limiting Cases:

Motion in a Vertical Circle:

  1. At the top, the centripetal force equals the the weight of the object. (The minimum force required to maintain the object in circular motion)

  1. Conservation of energy relates the velocity at the top and at the bottom.

  1. Tension forces at the top and bottom can be related.

 or

Rotational Motion

Three important equations:

Particle in Circular Motion:

With Constant Angular Acceleration:

Torque:

Conditions for Equilibrium:

Changing from Linear to Rotation:

Moment of Inertia (MoI):

HOW TO DERIVE MOMENT OF INERTIA (example):

Parallel Axis of Theorem:

Example)

The Moment of Inertia (REMAINING):

Combined:

Angular Momentum:

Work done by Torque:

Work-Kinetic Energy Theorem:

Power by Torque:

Rotation, Translation, Rolling:

                                                                                                


Oscillations and Gravitation

Oscillation

Restoring Force

ex) spring                                         pendulum                                  

a. Spring

Restoring force of spring= F= ma= -kx

Where x is displacement from equilibrium position and k is a spring constant (dependent on the spring)

Note that hooke’s law always has a negative sign.

→ This is due to the fact that restoring force always acts opposite to the displacement. Further away the object from equilibrium position, more force acting on the object in the opposite direction to go back to the equilibrium position.

Period of a Spring.

T= 2

Energy

We know that potential energy

u= -dx

Therefore, elastic potential energy (spring potential energy) is equal to

U = - dx

   = ½ kx2

b. Simple Pendulum

In the figure above, note that gravity is responsible for making the pendulum to go back to its equilibrium position (angle= 0o).

Period of Simple Pendulum

T=  2

*Can only be used when angle is small.

Energy

When the pendulum is at the highest point in any direction (left or right)...

  1. Its instantaneous velocity is 0, making kinetic energy KE= ½ mv2= 0.
  2. It has maximum potential energy of PE= mghmax

When the pendulum is at the lowest point (passing through the equilibrium position)

  1. It has maximum kinetic energy KE= ½ mvmax2
  2. Its height 0, making potential energy PE= mgh= 0

Gravitation

Universal Law Of Gravitation

Gravitational Field

Gravitational Field= Acceleration due to gravity

g= =

On earth, g= 9.81 m/s2

Kepler’s Law

 1st law: The orbit of a planet is an ellipse with the sun at one of the two foci.

2nd law: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

Therefore,

A1= A2

½ r1l1 = ½ r2l2

r1l1 = r2 l2

And then, dividing both by time t

r1= r2 

r1v1= r2v2

3rd law: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

T2 = a3

Therefore, T2 is proportional to a3.

Satellite

From circular motion, we learned that

Centripetal force= F = .

We know that satellite is experiencing universal law of gravitation and therefore moving in a circle.

Therefore,

=

Simplifying,

v=

Therefore, for the satellite to be constantly rotating around the Earth with constant distance r, its velocity must be v= .

*

Lower velocity will lead to satellite eventually reaching the Earth.

Higher velocity will lead to ecliptic path or even satellite completely moving away from the Earth.

Gravitational Potential Energy

We know that potential energy

u= -dx

Therefore, gravitational potential energy is

u = -dx

  =

*Note that gravitational energy is not equal to mgh.

PE=mgh is only used when the height is much smaller than the radius of the planet so that the object experiences almost constant gravitational field regardless of its height.

https://people.rit.edu/vjrnts/courses/matter/labs/hookeslaw/handout.html

https://phys.org/news/2015-02-law.html