Introduction To Dynamics SS2 Further Mathematics Lesson Note

Dynamics is a branch of physics concerned with the study of forces and torques with their effect on motion.

Forces: Force can be defined as pressure which can cause an object to move. It can be a push or a pull, which can cause an object to change direction.

TERMS USED IN DYNAMICS

1. Particle: Anybody whose dimension can be neglected is called a particle.
2. Mass: The mass of a body is the amount of matter contained in the body.
3. Momentum: Momentum of a particle is the product of mass of the particle and its velocity.

NEWTON’S LAWS

1. First Law Of Motion: Everybody continues in its state of rest or in uniform motion in a straight line, unless it is compelled to change that state by an external force. 

If we place a ball on the ground, it will continue to rest there until someone comes to kick it. Once it is kicked it will start moving and continue to move until something happens, either to stop it or change its direction of motion. Examples are motors and brakes.

2. The Second Law Of Motion

The law state that “the rate of change of momentum of a body is proportional to the applied force and is in direction of the force”

Mathematically; 𝐹 ∝ ( 𝑑 ÷ 𝑑𝑡) (𝑚𝑣)

𝐹 = 𝑘𝑚 (𝑑𝑣 / 𝑑𝑡)

Where k is a constant.

Since 𝑑𝑣/ 𝑑𝑡 = 𝑎 (𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛)

𝐹 = 𝑘𝑚𝑎

If 𝑘 = 1,

𝐹 = 𝑚𝑎

Where a is the acceleration of the body and m is a mass. The unit of force is the Newton (N)

3. Third Law Of Motion

Newton’s third law of motion states that “Action and reaction are equal and opposite”. When two bodies are in contact, the force of action and reaction are equal in magnitude and opposite in direction.

APPLICATION OF NEWTON’S LAWS OF MOTION

Example 1: A boy sits on a bench, the mass of the bench is 10kg and the weight of the boy is 50N. What is the reaction of the ground on the bench on which the body is sitting? (𝑔 = 9.8𝑚/𝑠)

Solution 

Weight of the bench 𝐹 = 𝑚𝑎 (2𝑛𝑑 𝑙𝑎𝑤) = 𝑚𝑔 = 10 × 9.8 = 98𝑁

Weight of the boy = 50𝑁 (given)

Therefore, Weight of the bench and boy = 98 + 50 = 148𝑁

By the third law of Newton, the ground will exert an equal but opposite force on the bench on which the boy is sitting

Therefore,

𝑅 = 148𝑁

Example 2: A particle of mass 5kg on a smooth surface is acted upon by a force F. If the acceleration 

produce in the particle is 4𝑚/𝑠² , find the magnitude of force F

Solution 

By Newton’s 2nd law of motion

𝐹 = 𝑚𝑎

𝑚 = 5𝑘𝑔, 𝑎 = 4𝑚/𝑠²

𝐹 = 5𝑘𝑔 × 4𝑚/𝑠² = 20𝑁

Example 3: A constant force F of magnitude 100N is applied to a body of mass 10kg. Find the acceleration produced in the body

Solution 

𝐹 = 𝑚𝑎 (2𝑛𝑑 𝑙𝑎𝑤)

100 = 10𝑎

𝑎 =100 / 10

= 10𝑚∕s²

ASSIGNMENT

1. What would be the acceleration of a mass of 50kg, when acted upon by a force of 30N?
2. A force F acts on a body of mass 5kg on a smooth horizontal floor. If it produces an acceleration of 4.5𝑚/𝑠², find the magnitude of force F
3. The speed of a car of mass 500kg is reduced from 20 m/s to 15m/s in 4 seconds, by applying the brakes. Calculate the braking force applied