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5. Using Newtons Laws

Using Newtons Laws

- General Method
- Determine the object, or objects, of interest.
- Determine real forces are acting on each object.
- For each object, find the net force.
- Insert the net force into the 2nd law and solve

Multiple Objects

Example Saving a Climber

Newtons 2nd law applies to each climber For

this example, we assume no friction rope does

not stretch rope of negligible mass

Example (2)

- Forces on climber Steve
- Gravity
- Normal force
- Tension in rope
- Forces on climber Paul
- Gravity
- Tension in rope

Example (3)

Steve

Paul

1. Choose coordinate system for

each climber. 2. Sum forces for each and apply

2nd law.

Example (4) Steve

Steve

Example (5) Paul

Paul

Note for Paul, we have chosen a frame of

reference with x pointing down.

Example (6)

As usual, we equate components. But for this

problem only the x components are relevant

Steve Paul

Example (7)

By assumption

Therefore,

Steve Paul

Example (8)

Acceleration

Rope tension

Circular Motion

Circular Motion

is a unit vector that points from the center of

the circle to the object.

is the velocity of the object. As the

object moves around the circle, the direction of

the velocity changes as does the direction of the

unit vector .

Circular Motion

Since the path is a circle, the velocity is

always perpendicular to the unit vector . We

can therefore write

where the unit vector is in the direction of

the velocity and perpendicular to the unit vector

.

Circular Motion

The unit vectors and are related by a

clockwise rotation by 90o. If we represent this

rotation by the symbol R we can write one unit

vector in terms of the other

and the velocity as

Circular Motion

By definition, the acceleration is

since the speed v and rotation R are constant.

Circular Motion

The change in the unit vector is in the same

direction as the velocity. Therefore, we can

write

where c is a constant to be determined.

Circular Motion

The constant c is the speed with which the tip

of the unit vector moves around the circle of

unit radius

Therefore,

Circular Motion

When

is substituted into

we get

since

Circular Motion

We see that the acceleration is directed towards

the center of the circle, that is, it is

centripetal, and its magnitude is

Example 5.7 Loop-the-Loop!

What is the minimum speed needed to

guarantee that a roller-coaster car stays on the

track at the top of the loop?

- Identify forces on car
- Gravity
- Normal force from track

Example 5.7 (2)

We have just seen that to move in a circle,

an object must have a centripetal

acceleration. According to the 2nd law, the

acceleration is caused by a net force.

Example 5.7 (3)

Since the acceleration is in the same direction

as the net force, it follows that the net force

must be centripetal, that is, directed towards

the center of the loop. What are these forces?

Example 5.7 (4)

Presumably, they must be the two forces we have

identified the weight and the normal force.

As usual, we need to set up a coordinate system.

Example 5.7 (5)

Coordinate system Take y to be downwards

Take x to the right

Example 5.7 (6)

At the top of the loop, the normal force and the

gravitational force point downwards and towards

the center of the circle. Therefore, in the y

direction

Example 5.7 (7)

Solving for v we get

At the minimum speed the car is on the verge of

leaving the tracks at the top of the loop. This

occurs when the normal force, n, is zero!

Friction

The Nature of Friction

- Friction is an electrical force between the

molecules of surfaces in contact. - Unlike gravity, however, friction is a very

complicated force to describe accurately.

The Nature of Friction

- But, for many everyday situations, such as

dragging an object along a floor, we can describe

frictional forces using simple, approximate, - expressions.

Frictional Forces

- Static Friction This is the frictional force

between surfaces that are at rest relative to

each other. The maximum static frictional force

is found to be - fs ms n
- where n is the magnitude of the normal force.

ms is called the coefficient of static friction.

Frictional Forces

- Kinetic Friction This is the frictional force

between surfaces that are moving relative to each

other. Its value is found to be - fk mk n
- where n is the magnitude of the normal force.

mk is called the coefficient of kinetic friction.

Frictional Forces

- It is found that as the applied force increases

so does the opposing frictional force until a

maximum value is reached. When the applied force

exceeds the maximum frictional force the object

accelerates. - During acceleration the frictional force

decreases and remains constant when the motion is

constant.

Frictional Forces

Maximum frictional force Accelerating

ms n

mk n

Frictional force

Constant speed

At rest

Time

Frictional force remains equal to and opposite

the applied force.

Friction in Action

Without friction it would be impossible to walk

or make a vehicle move.

As you push against the ground, the ground pushes

you forwards!

Example Dragging a Box

What rope tension is needed to move the box

at constant velocity, assuming a coefficient of

kinetic friction mk between box and floor?

Example Dragging a Box

Draw free-body diagram for box. The magnitude of

the kinetic friction force is fk mk n

Example Dragging a Box

The motion is constant, so the forces cancel

fk T cosq 0 (x-dir.) mg n T sinq

0 (y-dir.)

Example Dragging a Box

The magnitude of the tension is therefore

Summary

- Big Idea A net force causes changes in motion.
- How to Apply
- Find all real forces on a body, sum them, and

apply Newtons 2nd and 3rd laws. - Frictional force
- Increases until object moves, then reduces and

remains constant when motion is constant.