Topic 2 Mechanics

2.2 Forces and Dynamics

Force Diagrams

In order to solve dynamics problems we usually require a diagram.

In general there are two types of diagram we can use.System diagrams These are a realistic picture of the whole system.

Free body diagrams These show only one object and only the forces acting on it.

Both types of diagram are useful in different situations.

Free Body Diagrams

Consider a box, mass m, that is sliding down a rough ramp of angle 30o

The system diagram showing the external forces looks like this:W is the box's weight acting vertically downwards from the centre of mass.

F is the surface friction force acting along the surface from the point of contact.

F

W

Free Body Diagrams

Consider a box, mass m, that is sliding down a rough ramp of angle 30o

The free body diagram for the box looks like this:W is the box's weight acting vertically downwards from the centre of mass.

R is the normal reaction force of the ramp on the box acting perpendicular to the surface of contact from the point of contact.

F is the surface friction force acting along the surface from the point of contact.

F

R

W

Free Body Diagrams

Often the only way to properly describe all the forces acting on an object is to draw a free body force diagram.

This will help identify Newton's third law paired internal forces and help solve the problem.

Weight due to Gravity

Gravity is not a force.

Gravity is a field. The slope of that field is the gravitational field strength.

The effect of the field on an object's mass is called the object's weight due to gravity.

Weight is a force and is measured in newtons.

Mass is a measure of an object's linear inertia and is measured in kilograms.Inertia is the resistance of an object to change its state of motion.

Weight due to Gravity

Weight due to gravity is calculated by:W = mgW : weight in N

M : inetial mass in kg

g : gravitational field strength Nkg-1

For object's in a uniform gravitational field (i.e. close to the Earth's surface) g is a constant.

G = 9.81 Nkg-1 or 9.81 ms-2

Newton's First Law

An object in equilibrium will remain in equilibrium if no external forces act.OR

If the resultant force on an object is zero, then the velocity of that object will be constant.If the resultant forces on an object are zero then the object is in translational (linear) equilibrium.

The velocity is constant or zero

There is no change in direction.

Newton's First Law

All of the objects below are in equilibrium. Determine the magnitude and direction of the unknown force.

m=6kg40N

F

m=4kg10N

F

m=8kg25N

F

300

Newton's Second Law

An object that is not in equilibrium experiences an acceleration proportional to the nett force on the object.F = maF: the vector sum (nett) force acting in N

m : the inertial mass of the object in kg

a : the acceleration of the object in ms-2

This is only true if m remains constant.

The acceleration will be in the same direction as the nett force.

Newton's Second Law

Calculate the acceleration of each of the objects below:

m=6kg40N

m=8kg40N

m=4kg40N

30N

Linear Momentum

The linear momentum of an object is the product of its inertial mass and its velocity.It is a sort of measure of how difficult it is to stop the moving object.

p = mvp = momentum in kgms-1

m = inertial mass in kg

v = velocity in ms-1

Momentum is a vector quantity.It has direction

Momentum is a conserved quantity.There is always the same amount of momentum in a system if no external forces act.

Linear Momentum

Calculate the momentum of the following objects.A 70kg man running with a velocity of 10ms-1 to the right.

A 1000kg car travelling at 30ms-1 to the left.

A 50,000kg rocket travelling at 80ms-1 upwards.

Newton's Second Law

Newton's second law can also be stated as:The nett force is equal to the time rate of change of momentum.

Impulse

In most situations such as collisions, F is not a constant and is applied over a very short time t.

The product of force and time is called impulse and is measured in Ns

Impulse = Ft = p = m(v-u)It is the total impulse of a kick that affects the change in momentum of a ball, not the force or time individually.

The total impulse is found in reality by measuring the area under an F-t graph

Impulse

A model rocket motor has a total impulse of 246Ns. The rocket has a mass of 500g and is fired from rest. What is the rocket's velocity when the motor burns out?

A cricket ball has a velocity of 130kmh-1 when bowled. The batsman applies an impulse of 350Ns to the 160g ball. What is the speed of the ball after striking?

Principle of Conservation of Linear Momentum

The Principle of Conservation of Linear Momentum states that:In a collision the total momentum of the system is conserved if no external forces act.

i.e. pfinal = pinitial OR mv = mu

Simple Collisions

Consider a car of mass 1500kg travelling at 50kmh-1. It strikes a stationary van (no brakes) of mass 2500kg. After the collision both vehicles move off together. Calculate the speed.

Simple Collisions

Two marbles (A and B) of equal mass (100g) roll towards each other. A has a speed of 2.5ms-1. B has a speed of 4.0ms-1. After the collision both marbles roll away with the same speed. Calculate the speed.

Newton's Third Law

Newton stated that:If body A exerts a force on body B then body B exerts an equal and opposite force on body A.

This is a consequence of the law of conservation of momentum.The two objects in a collision, are subject to the same impulse during a collision for the same time. The forces must therefore be equal

These forces are internal to the system. You can only see them in free body diagrams.

You can not measure these forces, but you can calculate them.

Newton's Third Law

Forces due to Newton's third law always occur in pairs.The pair of forces act:On different bodies

Are of the same type (contact, gravitational etc)

Are of the same magnitude

Are of opposite directions.

e.g. Bob pushes to the left on the wall.The wall pushes to the right on Bob

e.g. the Moon is attracted by the gravitational pull of the Earth.The Earth is attracted by the gravitational pull of the Moon.

Newton's Third Law

Two boxes (A and B) have masses 4kg and 9kg respectively and are in contact on a smooth floor. A force of 10N is applied to the 4kg box to cause an acceleration. What is the acceleration of the 9kg box?

What is the magnitude of the force exerted on Box B by box A?