One-Dimensional Kinematics

- One-Dimensional Kinematics only moves in one direction

- Varies in x/y/z axis only

- Distance

- Distance is a scalar quantity: it is the total length of movement

- Displacement

- Displacement is a vector quantity: it is the change in position
- Displacement = change in position

= final position - initial position

- Speed

- Speed is a scalar quantity: it shows how fast an object is moving

- Velocity

- Velocity is vector quantity: it shows the speed and the direction of an object

- Acceleration

- Acceleration is a vector quantity that is defined as the rate at which an object changes its velocity

- Calculus

- Lots of calculus are involved in physics.
- Derivative of Displacement is Velocity

- s’(x)=v(x)

- Derivative of Velocity is Acceleration

- v’(x)=a(x)

- Antiderivative of Acceleration is Velocity

- ∫a(x)dx=v(x)

- Antiderivative of Velocity is Displacement

- ∫v(x)dx=s(x)

- SUVAT

- SUVAT are equations of motion in physics
- If the acceleration is constant, we can use SUVAT equations

- Acceleration has to be constant!

- We state displacement as S, initial velocity as U, final velocity as V, acceleration as A, and time as T.
- s=ut+(½)at2
- v=u+at
- v2=u2+2as
- s=[(u+v)/2]t
- s=vt-(½)at2.
- It is worth noting that each equation is missing one variable. Use different equations for different situations
- These equations are for linear motions. Similar equations will appear for angular motions as well.

Two-Dimensional Kinematics

- Two-Dimensional Kinematics moves in two direction

- It contains two side vectors
- Usually will move in x and y directions

- The characteristics are mostly the same as one-dimensional kinematics

- Distance, Speed, Displacement, Velocity, and Acceleration

- For the y-component, we (almost) always consider gravity

- Gravity is the force that attracts objects towards the center of the planet
- We replace ‘a’ for SUVAT to g or -g

- g is gravitational acceleration
- g=9.81m/s2
- The sign of g is determined by the direction of the force. If we set upwards (for the y-component) as positive, we replace it as -g. If we set downwards (for the y-component) as positive, we replace it as g

- Terminal velocity is the highest velocity attainable by an object as it falls through air

- There is no more acceleration for the y-component.

- To solve two-dimensional kinematics, we consider the two vector components separately.

- This is because one component can not affect the other component
- For instance, the fall time for a bullet and an apple is the same if they are dropped in the same location. Although bullet has a higher velocity for the x-direction, the force that is acting towards the ground is the same: it is gravity.

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

- Mass vs. Weight

- Weight ≠ Mass
- Mass of an object is the amount of matter in an object
- The weight of an object is the force of gravity affecting that object
- Weight=mass×(gravitational acceleration)=mg

- Normal Force

- Normal force is the force that is perpendicular to the surface of contact.
- It is usually represented as N.

- Newton’s First Law:

- Also called as the Law of Inertia
- An object will remain at rest or with a constant velocity unless it experiences an external net force
- It can be stated that if Fnet=0, then a=0

- Newton’s Second Law:

- Force is proportional to the rate of change in momentum
- Usually written as“Fnet=ma”

- One of the most important equations

- Newton’s Third Law:

- There is an equal and opposite reaction for every action there is

Work, Energy, and Power

- Work

- Work in physics is not the same as work we use in normal life
- Work is done on an object when a force causes a displacement.
- Work has an equation of

W=F⋅d=|F| |d| cosθ

- Where W is work, F is force, and d is displacement.

- Work has a unit of Joules (J)

- 1J = 1 N⋅m = 1

- Work is not done unless the force applied creates a change in the object’s location
- Constant force in 1D

- W = Fx ⋅ x

- Variable force in 1D

- W= ∫Fx ⋅ dx

- Constant Force in 3D

- W = F ⋅ r

- r contains all components (x, y, z)

- Variable force in 3D

- W = ∫F ⋅ dr

- Energy

- Energy is the ability to make things move or change
- Energy is defined as the ability to do work
- Energy has the unit of Joules (J)

- It has the same unit as work

- Energy can exist in many different forms, and it can change its form

- Kinetic Energy

- Kinetic Energy, or KE, is the energy of motion
- KE=(½)mv2

- KE = kinetic energy (J)
- m = mass (kg)
- v = velocity (m/s)

- Gravitational Potential Energy

- Gravitational Potential Energy, or PE, is the energy due to an object’s position
- “Zero potential energy” can be defined differently for different situations
- PE=mgh

- PE = gravitational potential energy (J)
- m = mass (kg)
- g = acceleration due to gravity = 9.81 (m/s2)
- h = height (m)

- Elastic Potential Energy

- Elastic potential energy, or PEs, is the energy due to compression or expansion of an elastic object such as a spring.

- Springs are mostly governed by Hooke’s Law

- Hooke’s Law state that restoring force is proportional to the extension
- Fspring = -kx

- F = force of the spring (N)
- k = spring constant (N/m)
- x = displacement (m)

- PEs=(½)kx2

- PEs = elastic potential energy (J)
- k = spring constant (N/m)
- x = displacement (m)

- Work and energy

- Using SUVAT and the formula for work, we can state that work is the change of kinetic energy

- W = (½)mv2 - (½)mu2 = ΔKE

- Mechanical Energy

- ME = KE + PEg + PEs

- Mechanical energy is a combination of KE and PE

- Conservation of Mechanical Energy states that the sum of KE and PE remains constant.

- One type of energy just changes into another type
- ME1 = ME2

- Power

- Power is how much time it takes to do a certain amount of work
- Watt is the unit of Power
- Power has an equation of P = ΔW/Δt = F ⋅ v

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?

- Circular motion is a movement of an object along the circumference of a circle or rotation along a circular path.

- It can be uniform or non-uniform.

- Examples include roundabouts in the playgrounds.

Uniform Circular Motion:

- An object in uniform circular motion has constant speed. However, it changes its direction (it’s constant speed not velocity) as it goes around the circle.

- The velocity vector is always tangent to the circle.

- An object is accelerating when the magnitude of the velocity changes or the direction of the velocity changes. Thus, an object in uniform circular motion is accelerating since the direction of the velocity vector is changing.
- The net force that is acting on an object in uniform circular motion is directed towards the center of the circle. This net force is called the centripetal force.

- This force is perpendicular to the velocity vector. With this inward force acting on the object, an inward acceleration called the centripetal acceleration results.

Centripetal Acceleration and Centripetal Force:

- Centripetal acceleration () formula can be derived (above) to be in m/s2.
- so centripetal force = in kg• m/s2.

Non-Uniform Circular Motion:

- An object in non-uniform circular motion would not have constant speed. In this case, there would be both tangential acceleration () as well as centripetal/radial acceleration.

- Whereas the centripetal acceleration makes the object to move in a circle, the tangential acceleration makes the speed changes.

- The net (total) acceleration, , is + , but the magnitude is.

Horizontal Circular Motion:

- Imagine that you are spinning mass with string attached on a table.
- The mass, which is in horizontal circular motion, don’t have any vertical as the normal force equals to .
- The tension force, which would towards your hand holding the string, would equal to the centripetal force.

Conical Pendulum:

- The mass around a circle in constant speed but the string describes a cone.

supports the weight.

Car on Horizontal Road:

- The vertical forces cancel out as the normal force equals to the weight of the car.
- IF the road is frictionless, circular motion is not possible. IF the road has friction, circular motion is possible.

Banked Frictionless Road & Limiting Cases:

Motion in a Vertical Circle:

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

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

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

or

Rotational Motion

- Like there is the kinematics to describe linear motion, there is the rotation to describe rotations.
- To turn linear motion into rotational motion...:

Three important equations:

Particle in Circular Motion:

- The position of the particle is in θ.
- The arc length (l) = rθ.

With Constant Angular Acceleration:

- Like SUVAT in kinematics, with constant angular acceleration, following equations can be used:

Torque:

- A torque is an influence which tends to change the rotational motion of an object.
- Torque (𝛕) = r ⨯ F = rF sin θ

Conditions for Equilibrium:

Changing from Linear to Rotation:

- The I is called moment of inertia.

Moment of Inertia (MoI):

- Moment of inertia is the resistance to a change in the rotation.

HOW TO DERIVE MOMENT OF INERTIA (example):

- ROD: The rod would be considered to be an infinite number of point masses. Thus, using the moment of inertia formula for point mass:

Parallel Axis of Theorem:

- The MoI of any object about any axis through its centre of mass is the minimum MoI for an axis in that direction in that space.
- The MoI about any axis parallel to that axis through CoM is:

- The expression added to the centre of mass MoI will be recognized as a point mass.

Example)

The Moment of Inertia (REMAINING):

Combined:

Angular Momentum:

Work done by Torque:

- because dr is just a portion of the circle travelled.

Work-Kinetic Energy Theorem:

Power by Torque:

Rotation, Translation, Rolling:

Oscillations and Gravitation

Oscillation

Restoring Force

- Force that allows a material to return to its original shape after being stretched or compressed.
- If the system is perturbed (moved) away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium.

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).

- Divide the mg vector into mgsin and mgcos.
- Note that if the pendulum is taut, the tension force of the pendulum FT would cancel out with mgcos .
- Therefore, only mgsin is responsible for restoring the pendulum.

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)...

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

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

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

Gravitation

Universal Law Of Gravitation

- m1= mass of one object, m2= mass of another, r= distance between two, G= gravitational constant
- Even small particles and huge planets follow this universal law.
- F1=F2, meaning that even if two particles have very different masses, the gravitational forces they exert on each other are equal in strength.

Gravitational Field

Gravitational Field= Acceleration due to gravity

g= =

On earth, g= 9.81 m/s2

- Because gravitational field is F divided by m, the gravitational field done by Earth is constant regardless of mass of any any object on earth.
- This is why acceleration due to gravity on Earth is constant regardless of mass.

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