Session 9 - Newton's Second Law

As we have seen, Newton's second law is easy to express mathematically:

F = m a

Strictly speaking, the force (F) is the NET FORCE. The mass (m) refers to the mass of the system, and the acceleration (a) also refers to the system.

The unit is the kg m / s^2

This is defined as a newton (N).

We consider a few cases in detail.

1. Weight

First, the force due to gravity - weight, W

Since F = m a, and the acceleration is due to gravity,

W = m g

Mass (m) is the amount of matter (or stuff). The weight is a way of quantifying how much gravity pulls on the mass. This means, if g is different, the weight is also different. You weigh less on the Moon, more on Jupiter, less at high altitudes, etc.

2. Inclined Planes

Objects resting on inclined planes are not free to fall directly down. Rather, they are constrained by the geometry of the plane. Part of the weight (a parallel component) can be thought of as acting DOWN the plane. Part of the weight (a perpendicular component) can be thought of as acting ONTO the plane. By trig:

W(parallel) = mg sin(theta)

W(perpendicular) = mg cos(theta)

Without any resistance whatsoever, all objects slide down a plane with the same acceleration:

F = m a

W(parallel) = m a

mg sin(theta) = m a

a = g sin(theta)

3. Friction

Friction is a catch-all term for any resistive force (really due to electromagnetic interactions between surfaces).

The frictional force (f) is the amount of force that resists motion - that is, it acts in the direction opposite the motion. We can quantify friction by introducing a coefficient of friction (u).

u = f / mg

Meaning: the ratio of frictional force that exists to the weight of the object is defined as the coefficient of friction. Typically, this is a (unitless) number much less than 1.

4. Circular Motion

As Newton's 1st law would predict, any object that moves in a circular path has a REASON to do that - some force is causing it to happen. Recall that acceleration is a change in velocity - since velocity refers to a magnitude (speed) AND a direction, if the direction of a body is changing, it MUST be accelerating even if the speed does not change.

Consider a ball spinning on a string at a constant speed. Even if the speed remains constant, we know that the ball is accelerating - its direction is constantly changing. Some force must be causing that to happen. We call such a force - a center-directed force - centripetal. The center-directed acceleration that results is called centripetal acceleration (ac).

ac = v^2 / r

Or, to compute the magnitude of the centripetal acceleration, we take the speed squared and divide it by the radius of orbit.

The units of acceleration are still m/s^2.