AP Physics C Mechanics Review

Vector

A quantity that involves both magnitude and direction

Scalar

A quantity that does not involve direction; Only has magnitude

Magnitude of a Vector

|A| or ||A||=√((Ax)^2+(Ay)^2)

Angle or Direction of a vector

θ=tan⁻¹(Ay/Ax)

Dot product

The multiplication of two vectors which results in a scalar. A·B=|A||B|cosθ=AxBx+AyBy

Cross product

The multiplication of two vectors which results in a vector.
A × B=<(AyBz-ByAz),(AzBx-BzAx),(AxBy-BxAy)>

Speed

v, distance over time, scalar, m/s

Velocity

v, displacement over time, vector, m/s

Equation for Velocity

v=∆s/∆t=ds/dt=∫a dt

Acceleration

a, change in velocity over time, m/s²

Equation for Acceleration

a=∆v/∆t=dv/dt=d²s/dt²

Kinematic equation without acceleration

∆x=x-x₀=vt

Kinematic equation without displacement

v=v₀+at

Kinematic equation without final velocity

x=x₀+v₀t+½at²

Kinematic equation without initial velocity

x=x₀+vt-½at²

Kinematic equation without time

v²=v₀²+2a(x-x₀)

Equation for position

s=∫v dt=∫(∫a dt)dt

Value of gravity on earth

g=9.80 m/s²≈10.0 m/s²

Formula for displacement of horizontal component projectile motion

∆x=v₀t (v₀ is the initial horizontal velocity)

Formula for final velocity of horizontal component projectile motion

v=v₀ (Note: velocity is constant in horizontal component of projectile motion!)

Formula for acceleration of horizontal component projectile motion

a=0

Formula for displacement of vertical component projectile motion

∆y=v₀t-½gt² (g is positive here! If you use a negative value for g then ∆y=v₀t+½gt²)

Formula for final velocity without displacement of vertical component projectile motion

v=v₀-gt (g is positive here! If you use a negative value for g then v=v₀+gt)

Formula for acceleration of vertical component projectile motion

a=-g (g is positive here! If you use a negative value for g then a=g)

Formula without time for final velocity of vertical component projectile motion

v²=v₀²-2g∆y (g is positive here! If you use a negative value for g then v²=v₀²+2g∆y)

Newton's First Law

The law of inertia: an object at rest stays at rest, an object in motion stays in motion unless acted upon by an outside force

Newton's Second Law

Fnet=ma

Newton's Third Law

Whenever two objects interact the force the first object exerts on the second object is equal to, but in the opposite direction, of the force the second object exerts on the first object.

Weight

Fw=mg

The normal force

The component of the contact force to the surface that is perpendicular to the surface. N or Fn

Friction

The component of the contact force exerted on an object in contact with the surface. It is parallel to the surface. Ff or f.

Formula for friction

F=µN

Coefficient of friction

The ratio of the force of friction between two bodies and the force pressing them together, µ.

Centripetal acceleration

Acceleration that towards the center of the circle

Formula for centripetal acceleration (uniform circular motion)

a=v²/r

Centripetal force

Any force, or component of a force, points toward the center of the circle if Fc>0. Any force, or component of force, points away from center of circle if Fc<0

Formula for centripetal force (uniform circular motion)

Fc=(mv²)/r

Force

A push or a pull, F, N.

Energy

The ability to do work, E, J.

Work

A way of transferring energy from one system to another, W, J.

The Law of Conservation of Energy

The first law of thermodynamics: The total amount of energy in any given process is conserved. Energy cannot be created nor destroyed it can only be transferred from one form to another.

Formulae for work

W=∫F·dr=F·x=∫F(x)dx=∆K=∆U=(Fcosθ)r

Kinetic Energy

Energy from motion, K, J.

Formula for kinetic energy

K=½mv²

Potential energy

Energy that is stored, mostly due to position, U, J.

Types of potential energy

Gravitational potential energy, Ug (due to gravity). Elastic potential energy, Uel.

Formula for elastic potential energy

Uel=½kx²

Formula for gravitational potential energy

Ug=mgh

Conservation of Mechanical Energy

The total mechincal energy of a system is constant when there are no non-conservative forces, (eg friction), acting on the system. Ei=Ef, Ki+Ui=Kf+Uf.

Differential definition of force

F=-dU/dx

Stable equilbrium

This occurs when the force restores the object back toward the equilibrium point after a disturbance

Unstable equilibrium

This occurs when the force moves the object further away from the equilibrium point after it is disturbed

The equilibrium point

This is the point where F=0.

Power

The rate at which work is done, P, W.

Formulae for power

P=W/t=dW/dt=Fv

Linear momentum

p=mv, kgm/s

Law of Conservation of linear momentum

total p before collision = total p after collision. Linear momentum is conserved unless acted upon by an outside force.

Elastic collision

A collision that conserves kinetic energy

Inelastic collision

A collision where kinetic energy is not conserved

Total or perfectly inelastic collision

This is a collision where the two objects stick together.

Impulse

Impulse is a change in momentum, J, Ns.

Formulae for Impulse

J=F∆t=∆p=∫F dt

Center of mass

The mean location of a distribution of mass.

Formulae for center of mass

r=∑mr/∑m=∫r dm (where dm=λdr).

Relationship between angle and position

s=rθ

Relationship between rotational velocity and linear velocity

v=rω

Relationship between rotational acceleration and linear acceleration

a=rα

Angle

The direction something as at, θ, rad or °.

Relationship between radians and degrees

π rad = 180°

Rotational velocity

The number of rotations something completes in an amount of time, ω, rad/s.

Rotational acceleration

The change rotational velocity, α, rad/s²

Formulae for rotational velocity

ω=v/r=dθ/dt=∫α dt

Formulae for rotational acceleration

α=a/r=d²θ/dt²=dω/dt

Formulae for angle

θ=s/r=∫ω dt= ∫(∫α dt) dt

Angular kinematics equation without angular acceleration

∆θ=θ-θ₀=ωt

Angular kinematics equation without angular displacement

ω=ω₀+αt

Angular kinematics equation without final angular velocity

θ=θ₀+ω₀t+½αt²

Angular kinematics equation without initial angular velocity

∆θ=ωt-½αt²

Angular kinematics equation without time

ω²=ω₀²+2α∆θ

Rotational Inertia

The measure of how hard it is to change an object's rotational motion, I, kgm²

Formulae for rotational inertia

I=mr²=∫r²dm

Parallel axis theorem

I=Icm+md²

Torque

A force's ability to cause an object to rotate, τ, Nm.

Formulae for torque

τ=r×F=F*(lever arm)

Newton's second law for rotation

∑τ=Iα

Formula for rotational kinetic energy

Krot=½Iω²

Formula for rolling kinetic energy

Krolling=Krot+ktrans=½Iω²+½mv²

Formulae for angular momentum

L=Iω=r×p

Conservation of angular momentum

Angular momentum is conserved unless acted upon by an outside torque.

Differential definition of angular momentum

∑τ=dL/dt

Static equilibrium for rotational motion

For an object to be in static eq. for rot. motion ∑τ=∑F=0.

Newton's law of gravitation

This gives the force between any two point masses regardless of their mass or location.

Formula for gravitational force

Fg=(G)(M₁)(M₂)/r²

General formula for gravitational potential energy

Ug=-(G)(M₁)(M₂)/r

Formula for gravitational centripetal acceleration

ag=GM₁/r²

Formulae for velocity of a circular orbit

v=2πr/T=√(GM₁/r)

Escape velocity

The velocity required to escape from the pull of an objects velocity

Formula for escape velocity

vesc=√(2GM₂/d) for M₁ to escape M₂'s pull

Formula between a small solid sphere inside the shell

Fg=GMmr/R³

Hooke's law

F=-kx