Introduction to Circular motion

Introduction to Circular motion

Introduction to Circular motion. Circular motion is motion in which a body moves in a circle about a fixed point.

For a body moving in a circle;

Its direction and velocity are constantly changing.

It has an acceleration called centripetal acceleration.

It has a force called Centripetal force acting towards the centre of the circular path.

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Note: When the object is released, it moves such that the direction of motion at any point is along a tangent to the circular path.

Forces acting on the body describing circular motion.

(i) Tension: Force acting towards the centre of the circular path. It provides the centripetal force.

(ii) Centripetal force: Force acting towards the centre of the circular path.

(iii) Centrifugal force: Force acting away from the centre of the circular path.

(iv) Weight: Force acting vertically down wards towards the centre of the earth.

Examples of circular motion
-Pendulum bob tied to a string whirled in a vertical or horizontal plane
-Planetary motion etc

Newton’s Law of motion

These are three laws that summarize the behavior of particles in motion.

Newton’s First Law of motion

Newton’s first law of motion states that a body continues in its state of rest or uniform motion in a straight line unless acted upon by an external force.

Inertia
Inertia is the reluctance of a body to move, when at rest or to stop when moving.
Thus, when a force acts on a body, the body;
Starts or stops moving.
Increases or reduces speed depending on the direction of the force.
Changes direction of motion.

Newton’s second law of motion
Newton’s second law states that the rate of change in momentum is directly proportional to the force acting on the body and takes place in the direction of the force.

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When we consider a force of 1N, mass of 1kg and acceleration of 1ms−2, then, k=1.Therefore;
𝐅= 𝐦𝐚

A newton; Is the force which acts on a mass of 1kg to produce an acceleration of 1ms−2.

Newton’s third law of motion
It states that action and reaction are equal but opposite.
When a body, A exerts a force on body B, body B also exerts an equal force in the opposite direction.

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The block exerts a weight, W= mg on the table and the table also exerts an equal reaction R on the block. R= mg, so that the net force on the block is zero and therefore there is no vertical motion.

Applications of Newton’s third law of motion

(a) Rockets and jets
Rockets and jet engines are designed to burn fuel in oxygen to produce large amounts of exhaust gases.
These gases are passed backwards through the exhaust pipes at high velocity (large momentum).

This in turn gives the Rocket or jet a high forward momentum which is equal but opposite to that of the exhaust gases.

𝒎𝒈𝒗𝒈=−𝒎𝑹𝒗𝑹

Where 𝑚𝑔𝑣𝑔 is the momentum of the exhaust gases, and 𝑚𝑅𝑣𝑅 is momentum of the Rocket.

(b) Motion in the lift
Consider a person of mass m standing in a lift, when the;

i) Lift is stationary or moving with uniform velocity

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ii) Lift is moving upwards with acceleration, a.

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iii) Lift is moving down wards with acceleration, a.

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Example:1
A person of mass 78kg is standing inside an electric lift. What is the apparent weight of the person if the;
d) Lift is moving upwards with an acceleration of 2ms-2?
e) Lift is descending with an acceleration of 2ms-2?

Solution
(a)

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𝐂𝐎𝐋𝐋𝐈𝐒𝐈𝐎𝐍𝐒 𝐀𝐍𝐃 𝐌𝐎𝐌𝐄𝐍𝐓𝐔𝐌

Linear Momentum:
Momentum is the product of mass and its velocity.

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The S.I unit of momentum and impulse is Kgms-1
Note: Momentum and impulse are vector quantities.

Principle of conservation of momentum
It states that when two or more bodies collide, the total momentum remains constant provided no external force is acting.

It states that when two or more bodies collide, the total momentum before collision is equal to the total momentum after collision.

Suppose a body of mass m1 moving with velocity u1 collides with another body of mass m2 moving with velocity u2. After collision, the bodies move with velocities v1 and v2 respectively, then;

𝒎𝟏𝒖𝟏+𝒎𝟐𝒖𝟐=𝒎𝟏𝒗𝟏+𝒎𝟐𝒗𝟐

Types of collisions

Elastic collision
Elastic collision is the type of collision whereby the colliding bodies separate immediately after the impact with each other and move with different velocities.

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Inelastic collision
Inelastic collision is when the colliding bodies stay together and move with the same velocity after collision.
In short, for inelastic collision,

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Comparisons between Elastic collision and Inelastic collision

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NOTE; For any stationary body or body at rest, the initial velocity is zero so the initial momentum of such a body before collision is zero.

Example:1
A body of mass 3kg traveling at 5ms-1 collides with a 2kg body moving at 8ms-1 in the same direction. If after collision the two bodies moved together, Calculate the velocity with which the two bodies move after collision.

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Example: 2
A body of mass 8kg traveling at 20 ms-1 collides with a stationary body and they both move with velocity of 15ms-1. Calculate the mass of the stationary body.

Solution

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Example: 3
A body of mass 20kg traveling at 5ms-1 collides with another stationary body of mass 10kg and they move separately in the same direction. If the velocity of the 20kg mass after collision was 3ms-1. Calculate the velocity with which the 10kg mass moves.

Solution

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