# Force, Mass, Acceleration and How to Understand Newton's Laws of Motion

## A Guide to Understanding Basic Mechanics

Mechanics is a branch of physics which deals with forces, mass, and motion.

In this easy to follow tutorial, you'll learn the absolute basics!

What's covered:

- Definitions of force, mass, velocity, acceleration, weight
- Vector diagrams
- Newton's three laws of motion and how an object behaves when a force is applied
- Action and reaction
- Friction
- Newton's equations of motion
- Adding and resolving vectors
- Moments, couples and torque
- Angular velocity and power

## Quantities Used in Mechanics

### Mass

This is a property of a body and a measure of an objects resistance to motion. It is constant and has the same value no matter where an object is located on Earth, on another planet or in space. Mass in the SI system is measured in kilograms (kg). The international system of units, abbreviated to SI from the French "Système International d'Unités," is the units system used for engineering and scientific calculations. It is basically a standardization of the metric system.

### Force

** **This can be thought of as a "push" or "pull." A force can be active or reactive.

### Velocity

This is the speed of a body in a given direction and is measured in metres per second (m/s).

### Acceleration** **

When a force is exerted on a mass, it accelerates. In other words, the velocity increases. This acceleration is greater for a greater force or for a smaller mass. Acceleration is measured in metres per second per second or metres per second squared (m/s^{2}).

### Load

When a force is exerted on a structure or other object, this is known as a load. Examples are the weight of a roof on the walls of a building, the force of wind on a roof, or the weight pulling down on the cable of a crane when hoisting.

Note: In US English, metres is spelled "meters"

## What Are Examples of Forces?

- When you lift something off the ground, your arm is exerting a force upwards on the object. This is an example of an active force

- The Earth's gravity pulls down on an object and this force is called
*weight* - A bulldozer can exert a huge force, pushing material along the ground

- A huge force or
*thrust*is produced by the engines of a rocket lifting it up into orbit

- When you push against a wall, the wall pushes back. If you try to compress a spring, the spring tries to expand. When you stand on the ground, it supports you. All these are examples of reactive forces. They don't exist without an active force. See (Newton's laws below)
- If the unlike poles of two magnets are brought together (N and S), the magnets will attract each other. However, if two like poles are moved close together (N and N or S and S), the magnets will repel

## What is a Newton?

Force in the SI system of units is measured in newtons (N). A force of 1 newton is equivalent to a weight of about 3.5 ounces or 100 grams.

## One Newton

## What Are Vector Diagrams?

In mechanics, vector or free-form diagrams are used to describe and sketch the forces in a system. A force is usually represented by an arrow and its direction of action is indicated by the direction of the arrowhead. Rectangles or circles can be used to represent masses.

## A Very Large Force

## What Types of Forces Are There?

### Effort

This can be thought of as the force applied to an object which may eventually cause it to move. For example when you push or pull a lever, slide a piece of furniture, turn a nut with a wrench or a bull dozer pushes a load of soil, the applied force is called an effort. When a vehicle is driven forwards by an engine, or carriages are pulled by a locomotive, the force which causes motion is known as the tractive effort. For rocket and jet engines, the term "thrust" is often used.

### Weight** **

This is the force exerted by gravity on an object. It depends on the mass of the object and varies slightly depending on where it is located on the planet and the distance from the center of the Earth. An object's weight is less on the Moon and this is why the Apollo astronauts seemed to bounce around a lot and could jump higher. However it could be greater on other planets. Weight is due to the gravitational force of attraction between two bodies. It is proportional to the mass of the bodies and inversely proportional to the square of the distance apart. In engineering, this force acting on a structure is known as a load.

### Tensile or Compressive Reaction

When you stretch a spring or pull on a rope, the material exerts an equal reactive force pulling back in the opposite direction. This is known as tension. If you try to compress an object such as a spring, sponge, gas or simply place an object on a table, the object pushes back. Working out the magnitude of these forces is important in engineering so that structures can be built with members which will withstand the forces involved, i.e they won't stretch and snap, or buckle under load.

### Static Friction

** ** Friction is a reactive force which opposes motion. Friction can have beneficial or detrimental consequences. When you try to push a piece of furniture along the floor, the force of friction pushes back and makes it difficult to slide the furniture. This is an example of a type of friction known as dry friction, static friction or stiction.

Friction can be beneficial. Without it everything would slide and we wouldn't be able to walk along a pavement without slipping. Tools or utensils with handles would slide out of our hands, nails would pull out of timber and brakes on vehicles would slip and not be of much use.

### Viscous Friction or Drag

** **When a parachutist moves through the air or a vehicle moves on land, friction due to air resistance, slows them down. If you try to move your hand through water, the water exerts a resistance and the quicker you move your hand, the greater the resistance. These reactive forces are known as viscous friction or drag.

### Electrostatic and Magnetic Forces

Electrically charged objects can attract or repel each other. Similarly like poles of a magnet will repel each other while opposite poles will attract. Electric forces are used in powder coating of metal and electric motors work on the principle of magnetic forces on electric conductors

## Weight is the Force on a Mass Due to Gravity

## What Are Newton's Three Laws of Motion?

**First Law**

"An object will continue in its state of rest or motion in a straight line provided no external force acts on it"

Basically, this means that if for instance a ball is lying on the ground, it will stay there. If you kick it into the air, it will keep moving. If there was no gravity, it would go on for ever. However, the external force, in this case, is gravity which causes the ball to follow a curve, reach a max altitude and fall back to the ground.

Another example is if you put your foot down on the gas and your car accelerates and reaches top speed. When you take your foot off the gas, the car slows down, The reason for this is that friction at the wheels and friction from the air surrounding the vehicle (known as drag) causes it to slow down. If these forces were magically removed, the car would stay moving forever.

### Second Law* *

"The acceleration of a body is directional proportional to the force which caused it and inversely proportional to the mass and takes place in the direction which the force acts"

This means that if you have an object and you push it, the acceleration is greater for a greater force. So a 400 horse power engine in a sports car is going to create loads of thrust and accelerate the car to top speed rapidly. Imagine if that engine was placed into a heavy train locomotive and could drive the wheels. Because the mass is now so large, the force creates much lower acceleration and the locomotive takes much longer to reach top speed.

If F is the force

m is the mass

and a is the acceleration

Then,

F = ma

Acceleration is measured in metres per second per second (or metres per second squared, abbreviated to m/s^{2})

**Example 1:** A force of 10 newtons is applied to a mass of 2 kilos. What is the acceleration?

F = ma

So a = F/m = 10 / 2 = 5 m/s

^{2}

The velocity increases by 5 m/s every second

### Weight as a Force

In this case, the acceleration is g, and is known as the acceleration due to gravity.

g is approximately 9.81 m/s^{2 }in the SI system of units.

Again F = ma

So if the force F is renamed as W, and substituting for F and a gives:

Weight

W = ma = mg

**Example 2:** What is the weight of a 10 kg mass?

W = mg = 10 x 9.81 = 98.1 newtons

### Third Law

"For every action there is an equal and opposite reaction"

This means that when a force is exerted on an object, the object pushes back.

Some examples:

- When you push on a spring, the spring exerts a force back on your hand. If you push against a wall, the wall pushes back.
- When you stand on the ground, the ground supports you and pushes back up. If you try to stand on water, the water cannot exert enough force and you sink.
- Foundations of buildings must be able to support the weight of the construction. Columns, arches, trusses and suspension cables of bridges must exert enough reactive compressive or tensile force to support the weight of the bridge and what it carries.
- When you try to slide a heavy piece of furniture along the floor, friction opposes your effort and makes it difficult to slide the object

## Test Yourself! - Quiz A

view quiz statistics## Examples of Active and Reactive Forces

## What is Dry Friction or "Stiction"?

As we saw above, friction is an example of a force. When you attempt to slide a piece of furniture along a floor, friction opposes your effort and makes things more difficult. Friction is an example of a *reactive* force, and doesn't exist until you push the object (which is the active force). Initially, the reaction balances the applied force i.e. your effort pushing the furniture, and there is no movement. Eventually, as you push harder, the friction force reaches a maximum, known as the *limiting force of friction*. Once this value is exceeded by the applied force, the furniture will start to slide and accelerate. The friction force is still pushing back and this is what makes it so difficult to continue to slide the object. This is why wheels, bearings, and lubrication come in useful as they reduce friction between surfaces, and replace it by friction at an axle and leverage to overcome this friction. Friction is still necessary to stop a wheel sliding, but it doesn't oppose motion. Friction is detrimental as it can cause overheating and wear in machines resulting in premature wear. So engine oil is important in vehicles and other machines, and moving parts need to be lubricated.

### Dry static friction, also known as stiction (see above diagram)

If F is the applied force on a body

The mass of the body is m

Weight of the body is W = mg

μ

_{s }is the coefficient of friction (low μ means the surfaces are slippery)and Rn is the normal reaction. (the reactive force at right angles to the surface due to the object being pushed against the surface)

Reaction Rn = Weight W

Then

limiting friction force is F

_{f}= μ_{s}R_{n }= μ_{s}W = μ_{s}mg

Remember this is the limiting force of friction just before sliding takes place. Before that, the friction force equals the applied force F trying to slide the surfaces along each other, and can be anything from 0 up to μR_{n}.

So the limiting friction is proportional to the weight of an object. This is intuitive since it is harder to get a heavy object sliding on a specific surface than a light object. The coefficient of friction μ depends on the surface. "Slippery" materials such as wet ice and Teflon have a low μ. Rough concrete and rubber have a high μ. Notice also that the limiting friction force is independent of the area of contact between surfaces (not always true in practice)

### Kinetic Friction

Once an object starts to move, the opposing friction force becomes less than the applied force. The friction coefficient in this case is μ_{k.}

## What Are Newton's Equations of Motion? (Kinematics Equations)

There are three basic equations which can be used to work out the distance traveled, time taken and final velocity of an accelerated object.

If u is the initial velocity

v is the final velocity

s is the distance covered

t is the time taken

and a is the acceleration

For uniform acceleration

v = u + at

s = ut + 1/2 at

^{2}v

^{2 }= u^{2}+ 2as

**Examples:**

*(*1) A force of 100 newtons accelerates a mass of 5 kg for 10 seconds. If the mass is initially at rest, calculate the final velocity.

Firstly it is necessary to calculate the acceleration.

F = 100 newtons

m = 5 kg

t =10 seconds

u = initial velocity = 0 since the object is at rest

F = ma so a = F/m = 100/5 = 20 m/s

^{2}

v = u + at = 0 + 20 x 10 = 200 m/s

(2) A mass of 10 kg is dropped from the top of a building which is 100 metres tall. How long does it take to reach the ground?

In this example, in theory it doesn't make any difference what the value for the mass is, the acceleration due to gravity is g irrespective of the mass. Galileo demonstrated this when he dropped two balls of equal sizes but differing masses from the leaning tower of Pisa. However in reality, the drag force due to air resistance will slow down the falling mass, so a 10 kg sheet of timber would fall slower than a 10 kg lead weight. So the calculations apply to a weight falling in a vacuum.

We know s = 100

g = 9.81

u = 0

We can use the equation s = ut + 1/2 at^{2}

u = 0 and a = g so s = 1/2gt

^{2}or t = √(2s/g) = √(2 x 100) / 9.81 = 4.5 seconds approx

## Commander David Scott's Hammmer and Feather Experiment from Apollo 15

## How to Add and Resolve Force Vectors

As mentioned briefly above, a force can be represented graphically by an arrow with a given direction known as a vector. If two or more forces are involved, problems in mechanics can be solved graphically by drawing the vectors, the head of one vector ending at the tail of the 2nd vector and so on. The vectors are drawn to scale, the length representing the magnitude of the force and the angle being the angle of action of the force. The "Triangle of Forces" or "Parallelogram of Forces" is then a method for visualizing or finding the resultant of forces.

Forces can also be resolved. In the diagram below, a mass rests on a slope. Using the parallelogram of forces in reverse, the weight force can be resolved into a force parallel to the slope and perpendicular to the slope. This is useful for these type of problems because it enables the normal reaction to be worked out (the force exerted by the slope on the mass as explained earlier) and also the friction forces involved.

## How to Use the Triangle of Forces to Add Vectors

## How to Use the Parallelogram of Forces to Add or Resolve Vectors

## Using the Parallelogram of Forces to Resolve Weight Into Normal and Tangential Forces

"Energy is the ability to do work"

## What is Kinetic Energy?

A body has kinetic energy when it is in motion.

If a body of mass m is moving at a velocity v, then:

Kinetic energy = (1/2)mv^{2}

**Example:**

A rifle bullet of mass 4 grams is moving at a velocity of 1200 m/s.

Calculate its energy.

First convert to SI units, so 4 g = 0.004 kg

Kinetic energy = (1/2)mv^{2 }= (1/2)0.004(1200)^{2} = 2880 joules

To put this into perspective, this is approximately the same impact energy that a 50kg / 8 stone weight would produce if dropped from chimney height off a two storey house. The bullet has a small mass, but the squared term in the equation increases the energy massively. Every time velocity is doubled, energy increases fourfold. If velocity is tripled, energy increases ninefold.

## What is Work?

The definition of work in physics is that "work is done when a force moves a body through a distance". If there is no movement of the point of application of a force, no work is done. So for instance, a crane that is simply holding a load at the end of its steel rope is not doing work. Once it starts hoisting the load, it is then doing work. When work is done there is energy transfer. In the crane example, mechanical energy is transferred from the crane to the load, which gains potential energy because of its height above the ground.

The units of work is the joule.

If work done is W

distance is s

and the force applied is F

then

W = FsCos Θ where Θ is the angle between the force and displacement

**Example:**

A force of 50 N is applied to a box of mass 4 kg resting on the ground. The force opposing motion is F_{f }= 2 N. Calculate the acceleration and work done sliding the box 3 m

Start by writing the force equation. The sum of forces produce a net acceleration:

F - F_{f} = ma

So substituting:

50 - 2 = 4 x a

Rearranging:

a = (50 - 2)/4 = 12 m/s^{2}

Work done on the box is the applied force multiplied by the distance. work is done opposing friction plus accelerating the box. The displacement is in the direction of the force, so Θ = 0 and Cos Θ = 1, so:

Work done = FCos Θs = 50 x 1 x 3 = 150 joules

## Engineering Mathematics by K.A. Stroud

This is an excellent math textbook for both engineering students and anyone with an interest in the subject. The material has been written for part 1 of BSc. Engineering Degrees and Higher National Diploma courses.

A wide range of topics are covered including matrices, vectors, complex numbers, calculus, calculus applications, differential equations, series, probability theory, and statistics. The text is written in the style of a personal tutor, guiding the reader through the content, posing questions, and encouraging them to provide the answer.

This book basically makes learning mathematics fun!

## What Are Moments, Couples and Torque?

When a force acts on an object, it produces what is known as a turning moment or just simply a moment. An example is when you push on the trunk of a small tree.This produces a turning moment about the base of the tree which is balanced by the tension in the trunk and the restraining force of the roots. If you push too hard, you exceed a breaking limit and the trunk snaps or the tree gets uprooted. The moment of a force about a point is the magnitude of the force multiplied by the perpendicular distance between the force and the point. When 2 forces act in opposite directions, this is known as a couple and the magnitude of the twisting force or couple is called torque. If the forces are both of magnitude F, and the perpendicular distance between them is d, then:

**Torque T = Fd**

As you can see, if the force is increased or the distance is increased, the torque becomes greater. So this is why it is easier to turn something if it has a larger diameter handle or knob. A tool such as a socket wrench with a longer handle has more torque.

A gearbox is a device which converts high-speed low torque to lower speed and higher torque (or vice versa). Gearboxes are used in vehicles to provide the initial high torque required to get a vehicle moving and accelerate it. Without a gearbox, a much higher powered engine with a resulting higher torque would be needed. Once the vehicle has reached cruising speed, lower torque is required (just sufficient to create the force required to overcome the force of drag and rolling friction at the road surface).

Gearboxes are used in a variety of other applications including power drills, cement mixers (low speed and high torque to turn the drum), food processors and windmills (converting low blade speed to high rotational speed in the generator)

A common misconception is that torque is equivalent to power and more torque equals more power. Remember however torque is a turning force and a gearbox which produces higher torque also reduces speed proportionately. So the power output from a gearbox is equal to the power in (actually a little less because of friction losses, mechanical energy being wasted as heat)

## Measurement of Angles in Degrees and Radians

Angles are measured in degrees, but sometimes to make the mathematics simpler and elegant it's better to use radians which is another way of denoting an angle. A radian is the angle subtended by an arc of length equal to the radius of the circle. Basically "subtended" is a fancy way of saying that if you draw a line from both ends of the arc to the centre of the circle, this produces an angle with magnitude of 1 radian.

An arc length r corresponds to an angle of 1 radian

So if the circumference of a circle is 2πr = 2π (r) the angle for a full circle is 2π

And 360 degrees = 2π radians

## Angular Velocity

Angular velocity is the speed of rotation of an object. Angular velocity in the "real world" is normally quoted in revolutions per minute (RPM), but it's easier to work with radians and angular velocity in radians per second so that the mathematical equations turn out simpler and more elegant. Angular velocity velocity denoted by the Greek letter ω is the angle in radians that an object rotates through per second.

## What is the Relationship Between Angular Velocity, Torque and Power?

If the angular velocity is ω

and torque is T

Then

Power = ωT

e.g.

A shaft from an engine drives a generator at 1000 RPM

The torque produced by the shaft is 1000 Nm

How much mechanical power does the shaft produce at the input to the generator?

1 RPM corresponds to a speed of 1/ 60 RPS (revs per second)

Each revolution corresponds to an angle of 2π radians

So 1 RPM = 2π/60 radians per second

And 1000 RPM = 1000 (2π/60) radians per second

So ω = 1000 (2π/60) = 200π/6 radians per second

Torque T = 1000 Nm

So power = ωT = 200π/6 x 1000 = 104.72 kW

## Questions & Answers

- Helpful 20
What is the magnitude of the friction between the wheels and the ground?

Friction is necessary between wheels and the ground in order to prevent the wheels slipping. Static friction doesn't oppose motion, but rolling friction can do so.

In the case of a wheel driving a vehicle, if the driving torque of the wheel turning clockwise is T and the wheel's radius is r, this results in a couple. So there's a force at the point of contact of the wheel and ground of F = T/r acting backwards and F = T/r acting forwards on the axle. If there's no slippage, a balancing force F = T/R acts forwards at the point of contact on the ground. So these forces are in balance. The other unbalanced force at the axle pushes the vehicle forwards.

Helpful 1A spring stretches by 6cm when supporting a load of 15N. By how much would it stretch when supporting a load of 5kg?

The extension is proportional to the tension in the spring (Hooke's Law)

So if F is the applied force , x is the extension and k is the spring constant

F = kx

or k = F/x

Plugging in the values

k = 15/6 N/cm

For a weight of 5 kg

F = mg

m = 5 kg

g = 9.81

So F = 5 x 9.81 = 49.05 N

Since F = kx for the spring

Rearranging:

x = F/k

Substituting values:

x = 49.05 / (15/6) = 19.62 cm

Helpful 1How do I calculate the magnitude of force when the amount of force is not given?

In that case, you would need info about acceleration/deceleration and mass and the time over which it occurs.

Helpful 16A ball is thrown vertically upward from the ground with a speed of 25.5m/s. How long does it take to reach its highest point?

My other article "Solving Projectile Motion Problems" deals with these sorts of problems. Check it out here:

Helpful 13

**© 2012 Eugene Brennan**

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