Chapter 5

Two Dimensional Forces

Equilibrium An object either at rest or moving

with a constant velocity is said to be in equilibrium

The net force acting on the object is zero (since the acceleration is zero)

0F

Equilibrium cont. Easier to work with the equation in

terms of its components:

0 0x yF and F

Inclined Planes Choose the

coordinate system with x along the incline and y perpendicular to the incline

Replace the force of gravity with its components

Forces of Friction When an object is in motion on a

surface or through a viscous medium, there will be a resistance to the motion This is due to the interactions

between the object and its environment

This is resistance is called friction

More About Friction Friction is proportional to the normal

force The force of static friction is generally

greater than the force of kinetic friction The coefficient of friction (µ) depends on

the surfaces in contact The direction of the frictional force is

opposite the direction of motion The coefficients of friction are nearly

independent of the area of contact

Applications Involving Friction, InclinesOn a microscopic scale, most surfaces are rough. The exact details are not yet known, but the force can be modeled in a simple way.

For kinetic – sliding – friction, we write:

is the coefficient of kinetic friction, and is different for every pair of surfaces.

Kinetic Friction, ƒk

The force of kinetic friction acts when the object is in motion

ƒk = µ n Variations of the

coefficient with speed will be ignored

4-8 Applications Involving Friction, Inclines

Static Friction, ƒs

Static friction acts to keep the object from moving

If F increases, so does ƒs

If F decreases, so does ƒs

ƒs µ n

The static frictional force increases as the applied force increases, until it reaches its

maximum. Then the object starts to move, and the kinetic frictional force takes over.

Applications Involving Friction, Inclines

Block on a Ramp, Example

Axes are rotated as usual on an incline

The direction of impending motion would be down the plane

Friction acts up the plane

Opposes the motion Apply Newton’s Laws

and solve equations

Atwood Machine

Atwood Machine Let’s take a look

at the forces on each mass…

Atwood Machine

Atwood Machine

Static Equilibrium Condition of an object when

net forces equal zero Object is motionless

mg

1

T1

2

T2 Since the sign is not accelerating in any direction, it’s in equilibrium. Since it’s not moving either, we call it Static Equilibrium.

Thus, red + green + black = 0.

Hanging sign f.b.d.

Free Body Diagram

T2

mg

T1

1 2

T1 cos 1

(x component)

T2 cos 2 (X component)

T1 sin 1

(y component)

T2 sin 2

(Y component)

T1 cos 1 = T2 cos 2

Horizontal:

Vertical:T1 sin 1 + T2 sin 2 = mg

Components & Scalar Equations

If in Equilibrium……..the following would be

true

Sample ProblemA mother and daughter are outside playing on the

swings. The mother pulls the daughter and swing (total mass 55.0 kg) back so that the swing makes an angle of 40.0° with the vertical (50.0 ° from horizontal)

A. 703N

What is the tension in each chain holding the swing seat and the daughter?

Sample Problem…

Is this a case of equilibrium?

Calculate the magnitude of the net force

45°40 °

150N160N

75N

Equilibrium Example – Free Body Diagrams