2.6.2 Force, Mass and Acceleration

Posted on April 16, 2020 by user

Newton's Second Law

The rate of change of momentum of a body is directly proportional to the resultant force acting on the body and is in the same direction.

Implication:

When there is resultant force acting on an object, the object will accelerate (moving faster, moving slower or change direction).

Formula of Force

From Newton's Second Law, we can derived the equation
(IMPORTANT: F Must be the net force)

Summary of Newton's 1st Law and 2nd Law

Newton's First Law:
When there is no net force acting on an object, the object is either stationary or move with constant speed in a straight line.

Newton's Second Law:
When there is a net force acting on an object, the object will accelerate.

Example:
A box of mass 150kg is placed on a horizontal floor with a smooth surface; find the acceleration of the box when a 300N force is acting on the box horizontally.

Answer:
F = ma
(300) = (150)a
a = 2 ms^-2

Example:
A object of mass 50kg is placed on a horizontal floor with a smooth surface. If the velocity of the object changes from stationary to 25.0 m/s in 5 seconds when is acted by a force, find the magnitude of the force that is acting?

Answer:
We know that we can find the magnitude of a force by using the formula F = ma. The mass m is already given in the question, but the acceleration is not give directly.

We can determine the acceleration from the formula

From the formula
F = ma = (50)(5) = 250N

The force acting on the box is 250N.

2.6.1 Force

Posted on April 16, 2020 by user

A force is push or pull exerted on an object.
Force is a vector quantity that has magnitude and direction.
The unit of force is Newton ( or kgms^-2).

Unbalanced Force/ Resultant Force

When the forces acting on an object are not balanced, there must be a net force acting on it. The net force is known as the unbalanced force or the resultant force.

When a force acts on an object, the effect can change the

size,
shape,
stationary state,
speed and
direction of the object.

Types of Forces

Friction

Friction is the force that oppose motion.
There are 2 types of friction that you need to know in SPM:

Static friction - friction between 2 surfaces that are not moving relative to each other
sliding friction - friction where an object slides or rubs against another surface.

Video below discuss the differences between these 2 types of friction.

Tension

When a rope attached to an object is pulling on the object, the rope exerts a force T on the object, and the magnitude T of that force is called the tension in the rope.

Weight (Gravitational Force)

1. Weight is the gravitational force acted on an object.
2. It is always point towards the centre of the earth.

Normal Reaction (Normal Contact Force)

Normal reaction is a force acting perpendicular to two surfaces in contact with each other.

Newton’s Third Law of Motion

Newton's third law of motion states that for every force, there is a reaction force with the same magnitude but in the opposite direction.

Examples

When a man standing on the surface of the earth, his body is pulled by the gravitational force from the earth.
According to Newton's Third Law of Motion, his body will also have another gravitational force that pull the earth towards him, with the same magnitude as the gravitational force of the earth, but in opposite direction.

(The earth pull the man, the man pull the earth.)

(The man push the wall, the wall push the man.)

(The block press on the table, the table push the block upward.)

2.5.4 Application of Conservation of Momentum

Posted on April 16, 2020 by user

Rocket

Mixture of hydrogen and oxygen fuels burn in the combustion chamber.
Hot gases are expelled through the exhausts at very high speed .
The high-speed hot gas produce a high momentum backwards.
By conservation of momentum, an equal and opposite momentum is produced and acted on the rocket, pushing the rocket upwards.

Jet Engine

Air is taken in from the front and is compressed by the compressor.
Fuel is injected and burnt with the compressed air in the combustion chamber.
The hot gas is forced through the engine to turn the turbine blade, which turns the compressor.
High-speed hot gases are ejected from the back with high momentum.
This produces an equal and opposite momentum to push the jet plane forward.

2.5.3 Explosion

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Before explosion both object stick together and at rest.	After collision, both object move at opposite direction.
Total Momentum before collision Is zero	Total Momentum after collision : m₁v₁ + m₂v₂
From the law of conservation of momentum: Total Momentum Before collision = Total Momentum after collision 0 = m₁v₁ + m₂v₂ m₁v₁ = - m₂v₂ (-ve sign means opposite direction)

Example:
A man fires a rifle which has mass of 2.5 kg. If the mass of the bullet is 10 g and it reaches a velocity of 250 m/s after shooting, what is the recoil velocity of the pistol?
Answer:
This is a typical question of explosion.

m₁ = 2.5 kg
m₂ = 0.01 kg
u₁ = 0 ms^-1
u₂ = 0 ms^-1
v₁ = ?
v₂ = 250 ms^-1

By using the equation of conservation of momentum principle
0 = m₁v₁ + m₂v₂
0 = (2.5)v₁ + (0.01)(250)
(2.5)v₁ = -2.5v₁ = -1 ms^-1

2.5.2 Conservation of Momentum

Posted on April 16, 2020 by user

Principle of Conservation of Momentum

The principle of conservation of momentum states that in a system make out of objects that react (collide or explode), the total momentum is constant if no external force is acted upon the system.

Sum of Momentum Before Reaction
= Sum of Momentum After Reaction

Formula

Example - Both Object are in the Same Direction before Collision
A Car A of mass 600 kg moving at 40 ms^-1 collides with a car B of mass 800 kg moving at 20 ms^-1 in the same direction. If car B moves forwards at 30 ms^-1 by the impact, what is the velocity, v, of the car A immediately after the crash?

Answer:
m₁ = 600kg
m₂ = 800kg
u₁ = 40 ms^-1
u₂ = 20 ms^-1
v₁ = ?
v₂ = 30 ms^-1

According to the principle of conservation of momentum,

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
(600)(40) + (800)(20) = (600)v₁ + (800)(30)
40000 = 600v₁ + 24000
600v₁ = 16000
v₁ = 26.67 ms^-1

Example - Both Object are in opposite direction Before Collision
A 0.50kg ball traveling at 6.0 ms^-1 collides head-on with a 1.0 kg ball moving in the opposite direction at a speed of 12.0 ms^-1. The 0.50kg ball moves backward at 14.0 ms^-1 after the collision. Find the velocity of the second ball after collision.

Answer:
m₁ = 0.5 kg
m₂ = 1.0 kg
u₁ = 6.0 ms^-1
u₂ = -12.0 ms^-1
v₁ = -14.0 ms^-1
v₂ = ?

(IMPORTANT: velocity is negative when the object move in opposite direction)

According to the principle of conservation of momentum,

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
(0.5)(6) + (1.0)(-12) = (0.5)(-14) + (1.0)v₂
-9 = - 7 + 1v₂
v₂ = -2 ms^-1

Elastic and Inelastic Collision

Elastic Collision

Elastic collision is the collision where the kinetic energy is conserved after the collision.

Total Kinetic Energy before Collision
= Total Kinetic Energy after Collision

Additional notes:

In an elastic collision, the 2 objects separated right after the collision, and
the momentum is conserved after the collision.
Total energy is conserved after the collision.

Inelastic Collision

Inelastic collision is the collision where the kinetic energy is not conserved after the collision.

Additional notes:

In a perfectly elastic collision, the 2 objects attach together after the collision, and
the momentum is also conserved after the collision.
Total energy is conserved after the collision.

Example - Perfectly Inelastic Collision:
A lorry of mass 8000kg is moving with a velocity of 30 ms^-1. The lorry is then accidentally collides with a car of mass 1500kg moving in the same direction with a velocity of 20 ms^-1. After the collision, both the vehicles attach together and move with a speed of velocity v. Find the value of v.

Answer:
(IMPORTANT: When 2 object attach together, they move with same speed.)

m₁ = 8000kg
m₂ = 1500kg
u₁ = 30 ms^-1
u₂ = 20 ms^-1
v₁ = v
v₂ = v

According to the principle of conservation of momentum,

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
(8,000)(30) + (1,500)(20) = (8,000)v+ (1,500)v
270,000 = 9500v
v = 28.42 ms^-1

2.5.1 Understanding Momentum

Posted on April 16, 2020 by user

Momentum

Momentum is defined as the product of mass and velocity.
Momentum is a vector quantity. It has both magnitude and direction.
The SI unit of momentum is kgms^-1

Formula:

Example:
A student releases a ball with mass of 2 kg from a height of 5 m from the ground. What would be the momentum of the ball just before it hits the ground?

Answer:
In order to find the momentum, we need to know the mass and the velocity of the ball right before it hits the ground.

It's given that the mass, m = 2kg.

The velocity is not given directly. However, we can determine the velocity, v, by using the linear equation of uniform acceleration.

This is a free falling motion,
The initial velocity, u = 0
The acceleration, a = gravirational acceleration, g = 10ms^-2
The dispacement, s = high = 50m.
The final velocity = ?

From the equation
v² = u² + 2as
v² = (0)² + 2(10)(5)
v = 10ms^-1
The momentum,
p = mv =(2)(10) = 20 kgms^-1

2.4.2 Applications of the Concept of Inertia

Posted on April 16, 2020 by user

When the handle of the hammer hit on the surface, the handle stop but the head of the hammer keep on moving downward, owing to the effect of inertia. As a result, hammer head will fit tighter into the hammer.

Japanese sumo wrestlers normally have great body mass. This is because the greater mass can increases the effect of inertia and hence more difficult to be moved by their opponents.

When a prey trying to run off from its predator, it will run in zig zag rather than a straight path.

Explanation:
A predator usually has greater mass and hence greater effect of inertia. It is more difficult for the predator to change direction compare to the prey.

Animals shake their body to dry their fur.
We can spin an umbrella to remove water on the umbrella.

Explanation:
When the direction (of the fur/ umbrella) change, the water droplets keep on moving forward due to the effect of inertia, and hence leave the fur of the animals/umbrella.

Ways to Reduce the Negative Effect of Inertia

The tank in a tanker lorry is divided into smaller tanks. This may reduce the motion of the liquid inside the tank and hence reduces the effect of inertia.

Seat belt and Airbag
The passengers in a car will be thrown forward when the car stop suddenly due to the effect of inertia. The seat belt and airbag can reduce the impulsive force acted on the body of the passenger and hence prevent serious injury.

2.4.1 Understanding Inertia

Posted on April 16, 2020 by user

Mass

Mass is defined as the amount of matter.
The SI unit of mass is kilogram (kg)
Mass is a scalar quantity.

Inertia

Inertia is the property of a body that tends to maintain its state of motion.

Newton's First Law

In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with a constant velocity (that is, with a constant speed in a straight line).

Jerking a Card

When the cardboard is jerked quickly, the coin will fall into the glass.

Explanation:

The inertia of the coin resists the change of its initial state, which is stationary.
As a result, the coin does not move with the cardboard and falls into the glass because of gravity.

Pulling a Book

When the book is pulled out, the books on top will fall downwards.
Explanation:

Inertia tries to oppose the change to the stationary situation, that is, when the book is pulled out, the books on top do not follow suit.

Pulling a Thread

1. Pull slowly - Thread A will snap.
Explanation:

Tension of thread A is higher than string B.
Tension at A = Weight of the load + Pulling Force

2. Yank quickly - Thread B will snap.
Explanation:

The inertia of the load prevents the force from being transmitted to thread A, hence causing thread B to snap.

Passengers in a Vehicle

Phenomenon 1

When a car stop, passengers in the car will be thrown forward.

Explanation
When a car stops, the effect of inertia causing the passengers' body maintain their motion forward. As a result, the passengers are thrown forward.

Phenomenon 2

When a car accelerates, passengers in the car will be thrown backward.

Explanation
When a car accelerates and moves faster, the body of the passenger tends to maintain its state of motion due to the effect of inertia. As a result, the passenger is thrown backward as he moves slower than the car.

Larger Mass - Greater Inertia

Bucket filled with sand is more difficult to be moved. It's also more difficult to be stopped from swinging.

Explanation:

Object with more mass offers a greater resistance to change from its state of motion.
Object with larger mass has larger inertia to resist the attempt to change the state of motion.

Empty Cart is Easier to be Moved

An empty cart is easier to be moved compare with a cart full with load. This is because a cart with larger mass has larger inertia to resist the attempt to change the state of motion.

Car with Higher Load is More Difficult to be Controlled

A car loaded with more passengers is more difficult to be controlled.

Explanation:
This is because, with more passenger, the mass increases, and hence increase the effect of inertia. This make the car more difficult to change its speed, more difficult to stop, and more difficult to change direction.

2.3.2 Velocity – Time Graph

Posted on April 16, 2020 by user

The gradient of the velocity-time gradient gives a value of the changing rate in velocity, which is the acceleration of the object.
The area below the velocity-time graph gives a value of the object's displacement.

Analysing Velocity-Time Graph

When analysing velocity-time graph, always keep in mind that the gradient of the graph is equal to the acceleration of the graph.
If the gradient is constant, then the acceleration is constant. If the gradient increase, then the acceleration increase and etc.
Table below shows a few cases of velocity-time graph for different types of motion.

Uniform velocity

Uniform acceleration

Increasing acceleration

Uniform deceleration

Decreasing acceleration

Converting a Velocity-Time graph to Acceleration-Time graph

In order to convert a velocity-time graph to acceleration time graph, we need to find the gradient of the velocity time graph and plot it in the acceleration-time graph.
The 5th and 6th videos below explain how to sketch an acceleration-time graph form a given velocity-time graph.

Free Falling

Free falling is a motion under gravitational force as the only force acting on the moving object.
The acceleration of a free falling object is always constant.
On the surface of the earth, the acceleration of is equal to 10ms^-2, and is named as gravitational acceleration.
In SPM, you need to know the graphs of free falling of the following movement
1. Launching object upward.
2. Dropping Object from High Place
3. Object Falling and Bounce Back

Launching Object Upward

When you launch an object upward, its velocity decreases at a constant rate, hence it's a straight line with negative gradient in a velocity-time graph.
Since the velocity decreases at a constant rate, hence the acceleration is constant. Also, the acceleration is negative because the speed decreases in positive direction. Therefore, the acceleration graph is a horizontal line in the negative domain.

Motion

Velocity-Time Graph

Acceleration-Time Graph

Dropping Object from a High Place

When an object drops from a high place, its velocity increases at a constant rate, hence it's a straight line with positive gradient in a velocity-time graph.
Since the velocity increases at a constant rate and the speed increases in positive direction hence the acceleration is constant and positive.

Motion

Velocity-Time Graph

Acceleration-Time Graph

Object Falling and Bounce Back

Motion

Velocity-Time Graph

Acceleration-Time Graph

2.3.1 Displacement-Time Graph

Posted on April 16, 2020 by user

There are 3 types of motion graph, namely
1. the displacement-time graph
2. the velocity-time graph
3. the acceleration-time graph.
When analysing a graph, it's important for us to know what's the physical quantity that's represented by the gradient of the graph and the area below the graph.
For example, in a displacement-time graph, the gradient represent the velocity of the moving object, whereas in a velocity-time graph, the gradient represent the acceleration of the moving object.
It's also important for you to know how to find the gradient of a straight line from a graph.
Sometime, you will be asked to convert a displacement-time graph to a velocity-time graph or convert a velocity-time graph to an acceleration-time graph.

Displacement - Time Graph

In a Displacement-Time Graph, the gradient of the graph is equal to the velocity of motion.

Analysing Displacement - Time Graph

When analysing displacement-time graph, always remember that the gradient of the graph represents the velocity of the graph.
Therefore, if the gradient of the graph is positive, the velocity is positive, and if the gradient of the graph is negative, the velocity is negative.
A negative velocity indicates that the object moves in opposite direction.
Table below shows the displacement-time graph of various motion.

Velocity = 0
This is a horizontal straight line, hence the gradient = 0.
Therefore, the velocity = gradient of the graph = 0, which means the object is stationary (does not move).

Uniform Velocity

The graph is a non-horizontal straight line, hence the gradient is not equal to 0. For a straight line, the gradient is constant, hence, the velocity of the moving object is uniform.

Negative Uniform Velocity
The graph is a non-horizontal straight line, with negative gradient. For straight line, the gradient must be constant. The negative value of gradient indicates that the object moves in opposite direction.
Therefore, this graph represents a motion with uniform velocity in opposite direction.

Increasing Velocity
The graph is a curve, shows that the gradient is not constant. The gradient increases over time, indicates that the velocity increases over time.

Decreasing Acceleration
The gradient decreases over time, shows that the velocity of the moving object decreases over time.