Dynamics

In this article, we will learn about forces and the addition of vectors as per the Secondary 3 Physics guidelines. We will study the types of forces and their effects. We will also learn in detail how to add the vectors which are in the same direction as well as the vectors that are at an angle to each other.

What are Forces?

A force is a push or a pull that one object exerts on another.

The SI unit for force is Newton (N).

Effects of Forces

A force can:               

• Produce motion
• Stop motion
• Cause an object to speed up
• Cause an object to slow down
• Change the direction of motion

Classification of Forces

A contact force is any force that occurs when two or more objects are touching each other. For example - pushing a car up a hill, pushing a ball across the room, etc.

The normal reaction force is a force that is exerted perpendicular to the contacting surfaces. It is a contact force.

Friction is a force that resists motion when the surface of one object comes in contact with the surface of another object. It is also a contact force and works only when the two objects are in contact and there is motion between them. 

Tension is a force that is transmitted through a cable, rope, wire or string when it is pulled tight by forces acting from opposite ends. It is always along a rope, cable, string or spring.

A non-contact force is a force applied to an object by another body that is not in direct contact with it. Non-contact forces come into play when objects do not have physical contact between them or when a force is applied without any interaction.

Gravitational force is the force applied to any object due to the Earth’s gravity. It is applied even when the object is not touching the ground. For example, an airplane flying in the sky will experience gravitational force due to the Earth’s gravity even though it is not touching the ground.

Electric force is a force of attraction or repulsion between any two charged bodies. This force also acts on the bodies that are not in contact with each other. For example, a positive charge will attract a negative charge even when they have some distance between them.

Magnetic force is a force of attraction or repulsion that is experienced by two bodies with magnetic poles. For example, the North pole of a magnet will attract the South pole of another magnet. Similarly, the North Pole of one magnet will repel the North Pole of another magnet.

Let us try the following question to understand the various forces better.

A box is pulled along a rough road towards the right. Label the forces.

Solution: 

In the above figure, we see that the object is being pulled with a rope on a rough road in the right direction.

So, the tension will be felt in the rope.

The weight of the body will exert a force on the road in the downward direction.

The normal reaction will be in the opposite direction to the weight of the object and will be perpendicular to the surface of the object and the road.

The friction force will be in the opposite direction to the direction in which the object is being pulled. The body will experience a frictional force as the surface of the road is rough. 

Practice Questions

Question 1:

Which of the following is not an effect of force?

1. It can change the shape of an object.
2. It can change the mass of an object.
3. It can change the direction of an object.
4. It can change the speed of an object.

Solution:

Option B is the correct answer.

Explanation:

1. The shape of an object can be changed by force. If we have a rubber ball in our hand and we squeeze it, its shape changes.
2. The mass of an object cannot be changed by any force. 
3. The direction of an object can be changed by applying a force. If the object is moving to the right and we apply a downward force, it can move in a diagonal direction.
4. The speed of an object can be changed by applying force. If the object is moving to the right and we apply the force from the left side to push the object to the right, it will move faster. Similarly, if we apply the force in the opposite direction to the movement of the object, the speed of the object will slow down.

Vector Diagram

Scalar quantities are physical quantities that have magnitude only.

To add scalar quantities, we add them numerically.

Vector quantities are physical quantities that have both magnitude and direction.

To add vector quantities, we add them using a vector diagram.

Vector Diagram

A vector diagram is usually made to scale. Most of the time we are asked to find the resultant vector diagrammatically, so we need to have the diagram made to scale in order to answer the question.

To represent a force of 20 N that acts at an angle of 45⁰.

Scale 1 cm: 5 N

Addition of Vectors

When there is more than one vector acting on an object, we must find a single vector that produces the same effect as the individual vectors combined.

This vector is known as the resultant vector.

A resultant vector is represented by a double-headed arrow.

Let us consider the following example.

Since the two vectors of 3 N each are in the same direction, we just add them up.

Resultant force = 3 + 3

                         = 6 N to the right

Mentioning the direction is very important as the vectors have magnitude as well as direction.

Let us consider another example.

The object has 3 N vectors in the right direction and 3 N in the left direction. So when we find the resultant vector, we subtract the vectors.

Resultant force = 3 - 3

                         = 0 N

Since the resultant is 0 N, we need not mention the direction.

Question 2:

Two forces act on an object as shown below. What is the resultant force acting on the object?

1. 1 N to the left
2. 1 N to the right
3. 7 N to the left
4. 7 N to the right

Solution:

Option B is the correct answer

Explanation:

Since the forces are in opposite directions, then the resultant force will always be in the direction of the larger force.

Resultant Force = 4 - 3 

              = 1 N to the right.

Addition of Vectors

When vectors are in the same direction or in opposite direction to each other, the resultant vector is found by adding or subtracting the individual vectors.

But how is the resultant vector found if the individual vectors are at an angle to each other?

1). Parallelogram Method

Let us consider the following example.  

The two vectors are acting at an angle of 45 degrees.  We make parallel lines to both the vectors using a ruler and set-square, completing a parallelogram. A diagonal is drawn from the starting point of the two vectors to the other end of the parallelogram. Then the length of the diagonal is measured. Using the scale, the magnitude of the resultant is found. 

Resultant Force = 9.2 × 1  

              = 9.2 N

Using a protractor to measure the angle between the horizontal force and the resultant force: 22.5⁰ from the horizontal force

Resultant force = 9.2 N at an angle of 22.5⁰ from the horizontal force

2). Tip To Tail Method

Using the above example, to use the tip-to-tail method, one of the given forces is selected to be drawn (to scale). The next force is then continued from the tip (head of arrow) of the first force. Geometry rules may have to be used to determine angles. 

The resultant will be from the tail of the first vector to the tip of the second vector. 

Force = 9.2 × 1 

          = 9.2 N at an angle of 22.5⁰ from the horizontal force

Forces in Equilibrium

An object is in equilibrium when the resultant force is zero i.e. all the forces are balanced.

When an object is in equilibrium, it is either at rest or moving at a constant velocity in a straight line.

Addition of Vectors

Example 1:

A 4 N force and a 5 N force act at an angle of 30⁰  to each other as shown in the diagram below. Using a graphical method, find the resultant force.

Solution:

Make sure that you choose an appropriate scale so that the diagram is big enough. The bigger the diagram, the lesser the error margin.

F = 8.7 cm x 1 N

   = 8.7 N at an angle of 16.7⁰ from the 4 N force

Addition of Vectors

Example 2:

An object of weight 10 N is hung from the ceiling. It is pulled to the left by a horizontal force of 6 N as shown in the diagram below. The object is in equilibrium. Determine the tension of the string.

Solution:

Scale 1 cm : 1 N

When the body is in equilibrium, then the arrows must be in a closed loop.

T = 11.7 cm x 1 N

   = 11.7 N at an angle of 30⁰ from the 10 N force.

Question 3:

Two forces act on an object as shown below. What is the resultant force acting on the object?

Solution: 

Using the Pythagorean theorem,

Resultant force = \(\sqrt{8² + 6²}\) 

      = \(\sqrt{64 +36}\)  

      = \(\sqrt{100}\)

              = 10 N

Question 4:

Three forces act on an object as shown below. What is the resultant force acting on the object?

Solution:

We will try to simplify the diagram by first simplifying the forces which are in the opposite direction.

We see that 6 N is downwards and 12 N is upwards.

So the net force of 6 N is in the upward direction. We have 8 N in the horizontal direction. We make the right angles triangle and find the resultant force.

Resultant Force = \(\sqrt{8² + 6²}\)

       = \(\sqrt{64 +36}\) 

       = \(\sqrt{100}\) 

       = 10 N

Question 5:

The diagram below shows two forces acting on an object.

Which of the following is their resultant force?

Solution:

Option 1 is the correct answer

Explanation:

Using the parallelogram method, we make the following diagram.

Option 1 is the correct answer.

Conclusion

In this article, we learnt about forces and the addition of vectors as per the Secondary 3 Physics level. 

• Forces
  • Contact forces and Non-contact forces
  • Effects of forces
• Addition of vectors
  • Vectors in the same or opposite direction
  • Vectors at an angle
  • Parallelogram Method of addition
  • Tip and Tail method of addition