Force and Motion

Girl pulls table cloth quickly but dishes remain on the table – a lot of physics is happening here

Inertia

According to Aristotle, the natural state of objects was to be at rest, and if you got them moving, eventually they would come to rest again. But being at rest is relative.

Galileo did experiments rolling balls down and up inclined planes, and realized that, in the absence of some kind of force, an object would keep moving forever once it got started.

Galileo called this phenomenon inertia

Inertial Frames

If you’re on a train moving 10 m/s, the world outside seems to be moving very fast. But if you look at objects inside the train, everything is still.

So, objects are at rest only in some frames. If a frame is not accelerating, it’s called an inertial frame. Root word: inert

If there are two trains, and the ground, there are many relative velocities. Velocities are relative! Consider that the earth is turning too…

Inertia

Inertia is the natural tendency of an object to maintain a state of rest or to remain in uniform motion in a straight line (constant velocity) in an inertial frame.

Mass is a measure of how much inertia an object has. Massive objects have more inertia, and it is more difficult to change the motion of a massive object.

Newton’s Laws of Motion

Newton’s 1st law of motion is sometimes called the law of inertia:

In the absence of any outside forces, a body at rest remains at rest, and a body already in motion remains in motion with a constant velocity (constant speed and direction).

Force and Net Force

A force is something that is capable of changing an object’s velocity.

Any particular force may not actually change an object’s velocity, because there may be other forces that prevent it from doing so.

But if the net force – the vector sum of all forces acting on the object – is not zero, the velocity of the object will change.

Force and Net Force

More about Forces

Different types of forces:

1.A contact force*, such as a push or pull, friction, tension from a rope or string, …

2.A force that acts at a distance, such as gravity, the magnetic force, or the electric force.

*Remember the 4 forces in nature – they are all non-contact forces. The electromagnetic force is the fundamental interaction responsible for contact forces. Everyday objects do not actually touch each other; rather contact forces are the result of the interactions of the electrons at or near the surfaces of the objects (exchange force).

Newton’s 2nd Law of Motion

Experiments show that the acceleration of an object is proportional to the force exerted on it and inversely proportional to its mass.

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The direction of the acceleration is in the direction of the applied net force.

F = m a

Newton’s 2nd Law of Motion

The units of force are called Newtons.

1 N = 1 kg.m/s2.

Newton’s 2nd Law of Motion

An object’s weight is the force exerted on it by gravity.

Here, g is the acceleration of gravity:

g = 9.81 m/s2

Weight therefore has the same units as force – Newtons

Newton’s 2nd Law of Motion

Newton’s second law applies separately to each component of the force:

Newton’s 3rd Law of Motion

For every force (action), there is an equal and opposite force (reaction).

Note that the action and reaction forces act on different objects.

This image shows how a block exerts a downward force on a table; the table exerts an equal and opposite force on the block, called the normal force N.

Newton’s 3rd Law of Motion

This figure illustrates the action-reaction forces for a person carrying a briefcase. Is there a reaction force in (b)? If so, what is it?

So, Newton’s Laws of motion

I. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

II. The relationship between an object's mass m, its acceleration a, and the applied force F is: F = ma.

III. For every action there is an equal and opposite reaction.

Free-Body Diagrams

A free-body diagram draws the forces on an object as though they all act at a given point.

You should draw such a diagram whenever you are solving problems.

Free-Body Diagrams

If an object has no acceleration, there must be no net force on it. Or, said in a different way, when there is no net force on an object, its acceleration is 0

This translates into three separate requirements: – that there be no force in the x direction, the y direction, or the z direction.

Always draw a free body diagram with all the forces acting on an object in order to analyze the situation

Free-Body Diagram: Simple Example

Ball hanging on a string

Draw all Forces acting on ball

mg

T

In x direction?

In y direction?

There is no NET force on the ball, so there is no acceleration

Free-Body DiagramsExample: pendulum from our first Lab:

Drawing free-body diagram:

there are two forces on the ball: Tension (T) and gravity (mg).

When the angle is not 0, is there a net force acting on the ball?

What about when is 0?

Free-Body DiagramsExample: pendulum from our first Lab:

y’

x’

First, change coordinate system to make analysis easier

Often, choosing coordinates where some of the forces are along coordinate axis simplifies the math of a problem considerably…

Free-Body DiagramsExample: pendulum from our first Lab:

y’

x’

Change coordinate system to make analysis easier

In y’ direction, ball cannot move, so acceleration in y’ is 0:

mg

co

s()

Fy = T - mg cos() = 0

mg sin()

Fx = mg sin() ≠ 0

Free-Body DiagramsExample: pendulum from our first Lab:

y’

x’

So in the picture, there is an acceleration along x’, and the ball moves in that direction

mg

co

s()

mg sin()

Fx = mg sin() = ma

a = g sin()

Free-Body DiagramsWhen the mass/string system is directly vertical, there is still an acceleration acting on the mass. Why? In what direction?

If the mass starts from θ = 10°, and the string length is L, can we calculate the maximum speed it has (when passing through the bottom of the motion?)

Yes, consider energy: ΔU = mgΔh, at the bottom the KE = ½ m v2, so

Free-Body Diagrams

so

y’

x’

Δh = L(1 - cos [] )

Period of a pendulumWe learned the equation of motion in x and y are completely separate. Consider a “conical pendulum”: it can be arranged to provide perfectly circular motion for a mass, where the mass never changes its vertical level

Period of a pendulumCircular motion is described by sines and cosines. If we look at the thing from the side, we see the mass going from left to right and back again

The mass goes around the circle with vmax, and the period is just the time it takes to go around: T = 2πR/vmaxR = L sin[]

Period of a pendulumThe period is

Here an approximation was made: when the angle is small, the term involving the angle is very nearly equal to 1….