Newton's 0

The terminal velocity of an object in free fall occurs when

the force on the object is 0

• frictional force cancels the gravitational force

• this is the maximum speed the object attains

Newton’s 3 laws of motion describe

how an object responds to interactions with other objects that exert forces on it

If the net force on the object is 0, it will have

0 acceleration

A nonzero force produces acceleration in proportion to

the mass of the object

The weight of an object is

the gravitational force exerted on it by Earth

If object A exerts a force on object B, then B

exerts an equal and opposite force on A

We don’t tilt our coordinate system, if the acceleration is

horizontal

For many problems, each of the forces that had to be resolved shows up twice — once with sine and once with a cosine factor. A set of force equations without matched sine and cosine factors is likely

incorrect

Make sure you don’t neglect any

equations

for centripetal acceleration problems, the maximum speed corresponds to

the maximum frictional force

When do you use the procedure for motion with resistive forces?

Situations where the force on an object (thus its acceleration) depends on its velocity

• Ex. air resistance, fluid resistance

Applying 2nd law gives you a differential equation to solve by integrating.

Motion with resistive forces, Step 1

Draw a free-body diagram

Motion with resistive forces, Step 2

Choose a coordinate system, one axis parallel to the net force if possible

Motion with resistive forces, step 3

Resolve all forces in the free-body diagram into components along the axes of coordinate system, apply Newton’s 2nd law

Motion with resistive forces, step 4

express all variables in your equations in terms of velocity and time to obtain a 1st order differential equation.

• acc. = 1st derivative of velocity - dv/dt

  • Since force depends on velocity, you must solve using calculus

Motion with resistive forces, step 5

Solve the differential equation by separating the variables and integrating

• Definite integrals are slightly easier to work with (v₀, v_f, t₀, t_f)

Motion with resistive forces, step 6

Use the final condition (behavior as time approaches infinity) to check your work.

• You should be able to do this w/o calculus

General Approach to Solving Newton’s Law Problems, Step 1

Draw a free-body diagram for each object

• Sketch that indicates all external forces exerted on the object

• If we’re talking motion of several objects, draw a separate diagram for each

General Approach to Solving Newton’s Law Problems, Step 2

Choose a coordinate system for each object

• Technically you could choose any (as long as it’s an inertial reference frame)

• Good choice simplifies problems

What coordinate system should you choose if it’s a linear motion problem?

• Choose one axis parallel to the direction of motion

• Ex. one axis should be parallel to the incline

What coordinate system is appropriate for UCM?

• One axis parallel to the radial coordinate, along the direction of centripetal acceleration

If there is no net force or acceleration, what coordinate system should you choose?

• Choose which axes are most convenient

• Try to align 1 axis parallel to the direction of possible motion

  • ex. one axis parallel to incline

If you’re solving a problem that contains multiple bodies with the same acceleration (e.g. connected by a rope),

make sure the accelerations of the objects are related appropriately

• positive acc. of one object in its coordinate system corresponds to positive acc. of a connected object in its coordinate system

General Approach to Solving Newton’s Law Problems, Step 3

Resolve all forces in the free-body diagram into components along the axes of your coordinate system.

• Add the components in each direction separately, applying Newton’s 2nd law

General Approach to Solving Newton’s Law Problems, Step 4

Solve the simultaneous equations from Newton’s 2nd law

Static equilibrium (object at rest) →

F_net,x = F_net,y = 0

(corollary of Newton’s first law)

Circular motion — solving equations

The centripetal force = net force required for circular motion

• Can be provided by tension, normal force, gravity, friction, etc.

Centripetal force should never appear on a free-body diagram, but it equals the net force when solving 2nd law problems.

In any UCM problem, the net force must equal

the centripetal force, F_net = mv²/r

net force must point toward the center of the circle

For solving equations,

you may also need to use equations that define the magnitudes of forces

• ex. equations for static & kinetic friction or gravity

the accelerations of any 2 objects attached by a string of fixed length have equal magnitudes (assuming string is massless and taut)

Why is it useful to designate θ and θ’?

keeping track of angles

• θ' = 90 - θ

  • complement of θ

if θ is one angle in a right triangle, the other must be θ’

if θ divides a right angle, the other angle must be θ’

• You can easily identify one angle after you’ve solved another

• quickly label all angles as either θ or θ’

• sin θ = cos θ’, and sin θ’ = cos θ

How do you calculate the coefficient of static friction?

Using the following equation,

• F_{maximum static friction} = μ_sF_N

Make sure you are familiar with the free-body diagram and geometry used in what types of problems that are commonly seen in the AP?

inclined plane problems

For Atwood machines, what are possible equations for the 2 masses?

F_net,m1 = F_T - m₁g = m₁a₁

F_net,m2 = F_T - m₂g = m₂a₂

v₁ = -v₂

a₁ = -a₂

For the pulley in the Atwood machine, what’s the upward force the pin must exert?

For the pulley not to fall down, it must exert an upward force exactly canceling the downward tension forces.

• Since the pulley is rotating, not translating, the net force is still 0.

F_net,y = F_pin - 2F_T

When several objects move together and accelerate at the same rate,

we can treat them as a single object to find their common acceleration.

To calculate the net force on a mass,

most cases you can simply substitute into Newton’s 2nd law.

For contact forces,

use Newton’s 3rd law. You can try finding the force one exerts on the other via a free-body diagram and applying the 2nd law.

In an elevator,

F_net,y = F_N - mg = ma

The scale reads the normal force.

Newton’s 1st Law

F_net = 0 → a = 0

n individual forces =

force = sum of

components, i.e., F = F_xi + F_yj

In problem solving, we add forces

component-wise, to get a set of 1D equations

• that can be related to 1D equations for UAM

Breaking F_net into components:

inertia

how much an object resists change in velocity

• measured by mass

v = 0 and/or v = constant is related to

a = 0 and F_net = 0

This is the basis of solving static equilibrium problems involving particles

What’s the difference between static equilibrium and dynamic equilibrium?

Static equilibrium = object at rest

Dynamic equilibrium = motion w/ constant velocity

Both involve 0 acceleration and 0 net force.

Newton’s laws are only valid in

inertial reference frames

• reference frames that move @ constant velocity w/ respect to other inertial reference frames

Example of an inertial reference frame

Earth (ignoring spin & revolution)

Noninertial reference frames

• attached to objects w/ linear acceleration (car example)

• centripetal acceleration (merry go round)

• linear & radial acceleration

Newton’s 2nd Law

F_net = ma

Newton’s 2nd Law implies force is a vector parallel to

acceleration

Force is measured in

lbs / Newtons

• 1 N = 1 kg m/s²

Newton’s 3rd Law

If a force is exerted by one object on another, another force equal & opposite is exerted back at the other object

Newton’s 3rd Law applies to

pair of objects

mass

measure of inertia

• SI unit = kg

weight

magnitude of force exerted on object by closest nearby planet

• w = mg

  • units of force (Newton / pounds)

magnitude of gravitational force

F = w = mg

Direction of gravitational force

towards center of the earth

• not necessarily perpendicular to surface

magnitude of normal force (F_N)

determined by 2nd law

direction of normal force

always perpendicular to surface object is on

pointing away from surface towards object

normal force characteristics

• equal & opposite, increases so net force = 0

• must have magnitude mg if a = 0

magnitude of frictional force

derived from experiments, it’s a function of applied horizontal force

static friction

F_fr = F_a

• whatever it takes to prevent motion

• objects aren’t sliding relative to each other

maximum magnitude of static friction

μ_sF_N

• μ_s = coefficient of static friction, property of 2 materials

kinetic friction

when objects are sliding

• magnitude = constant value = μ_kF_N

  • μ_k = coefficient of kinetic friction

the magnitude of kinetic friction is __ than the maximum static friction force

lower

F_static inequality

F_static ≤ μ_sF_N

F_kinetic equation

F_kinetic = μ_kF_N

direction of frictional force

it points parallel to the plane of contact

• its direction opposes the object’s motion

the normal force is perpendicular to

surfaces of contact

for friction, think

opposite of direction of motion

the minimum force for an object to move is

F_applied = μ_sF_N = μ mg

(given that vertical acceleration = 0, so it should just cancel out gravitational force (mg))

Once an object moves, if +x is the direction of motion, we’re only applying minimum horizontal force,

F_net,x = F_applied,x + F_friction,x

F_net,x = μ_smg - μ_kmg

a_net,x = F_net,x / m = (μ_s - μ_k)g

tension force - magnitude

determined by 2nd law

• ropes are approximated as massless so tension @ every point is the same

direction of tension force

parallel to rope / string

points towards middle of rope / string @ ends connected to objects

In Newton’s Law problems, be careful with the normal force, since

there are forces going against it you may forget to take into account.

For problems with graphs, give an example of why you should pay attention to details.

Asking questions, like “does velocity start at zero for this graph?” can quickly eliminate options.

A good idea is to familiarize yourself with ____ graphs.

position, velocity, and acceleration graphs (relative to each other)

Average force equation

Assuming mass is constant, based on F = ma, the average force = mass * average acceleration, (v_f - v₀)/Δt