Dynamics

Aslevels Physics Notes

Topic 4: Dynamics

One (of many) modern statements of the Newton’s laws of motion is as follows:

• 1st Law A body remains at rest or continues to move in a straight line with a uniform velocity unless a net external force acts on it. 
• 2 nd Law the rate of change of momentum of a body is proportional to the net force producing it, and takes place in the direction of this net force. 
• 3 rd Law the forces of action and reaction between interacting are equal in magnitude, opposite in direction and act on different bodies. 
  

• Newton’s 1st law (often called the law of inertia) suggests that matter has a property called inertia. 
• Inertia is the resistance to acceleration (or change in state of motion) upon the application of a force.
   e.g. a 3 kg mass has a greater resistance to acceleration than a 2 kg mass when they are under the action of the same net force. 
• The mass of a body remains constant regardless of its location. 
• Newton’s 1st law also suggests a quantity called force, which when acting on a mass causes its motion to change (in the direction of the force). 
• Weight is the effect of a gravitational field on mass. Its SI unit is the newton (N).
  It is a vector quantity. Its direction is the direction of the gravitational field strength. The weight of a body near the surface of the Earth refers to the force exerted on it by the gravitational attraction of the Earth.
  The weight W (in N) and the mass m i(in Kg) of a body are related by the expression;

Where g is the acceleration of the free fall due to gravitational attraction, in ms-²

• Unlike mass, the weight of a body may vary from place to place. 
• The mass m has appeared in two different situations. 

1. In Newton’s laws of motion, mass is a measure of the resistance to acceleration. It is usually called the inertial mass mi. 
2. In the concept of weight, mass is a measure of the amount of matter to be accelerated. It is usually called the gravitational mass m. 

Momentum

• In earlier analysis of motion, it was observed that the quantity (mass x velocity mad a fitting description for the amount of “motion”. A massive body was observed to have more “motion” than a less massive body moving at the same velocity. However, the less massive body can have equal or greater “motion if it is moving with a higher velocity. This quantity (mass x velocity) has been given a special name called  "momentum"

The momentum of a body is a vector quantity, its direction being the direction of the velocity. The unit of momentum in SI units is 𝑘𝑔𝑚𝑠 -¹ 𝑜𝑟 𝑁𝑠 there is no special name for this unit in SI system. Momentum is due to change in velocity as well as change in mass.

Although momentum is related to velocity, it gives no indication of how fast a body is moving. A large mass moving at low velocity can have the same momentum as a small mass moving at high velocity. A mass of 1000 kg moving at a velocity of 0.2 ms-¹ as a mass of 0.1 kg moving at a velocity of 2000 ms-¹ .

• In accordance with Newton’s 2nd law, Force is the rate of change of momentum 

Hence force is the rate of change of momentum. If a body is acted upon by a constant net force F of 2.0 N, its momentum p will change by 2.0 kgms-1 per second.

• When evaluating the change in momentum of a body, it is critical to use the net or resultant force, i.e. the vector sum of all forces acting on the body.  

Conservation of momentum

If no external force acts on a system of particles in a particular direction, then there can be no change in the total momentum of the system in that direction.
Suppose that two bodies of masses 𝑚₁ and 𝑚₂ are moving with velocities u₁ and u₂ respectively before collision

Since the force is defined as the rate of change of momentum i.e.

Collision

  A collision is a phenomenon which has these features:

•  It occurs in a short time interval.
• What happens after the collision differs from what happens before the collision  Colliding bodies may be assumed to constitute a closed system. 
• Momentum and energy are conserved during the collision.

 Head-on collision

In a head-on collision, the velocities of the colliding bodies are co-linear or directed along the same straight line both before and after the collision. Such collisions are also called collisions in one dimension.

Kinetic energy in collisions

In all collisions, momentum is always conserved.
However, in most collisions, some kinetic energy is usually lost.
The lost energy is usually converted into heat because of friction or work done in causing inelastic deformation to one or more of the colliding bodies.
Collisions are often classified according to whether the total kinetic energy of the system is conserved.

•  A (perfectly) elastic collision is one in which the total kinetic energy of the colliding bodies is conserved. 
• An inelastic collision is one in which the total kinetic energy of the colliding bodies is not conserved.

Most real collisions are inelastic, i.e. in most real collision, some changes in the total kinetic energy of the colliding bodies will occur.

•  Consider the elastic head-on collision of 2 bodies of masses m1 and m2 moving with velocities u₁ and u₂ before collision, and velocities v₁ and v₂ after collision, Kinetic energy is conserved  

i.e. total kinetic energy before collision =total kinetic energy after collision,

It means that the bowling ball will continue to move with the same speed even after collision.

 That means, the ping pong ball will move with a speed twice that of the bowling ball.

Then after collision. 𝑣₁ = 𝑢₁ And 𝑣₂ = 2𝑢₁

That is, for the large mass 𝑚₁, its velocity remains almost unchanged and the small mass 𝑚₂ moves off with a velocity twice that of the large mass. 

Case 2

 if 𝑚₂ is very large compared to 𝑚₁ (𝑚₂ ≫ 𝑚₁) and 𝑚₂ is at rest i.e. 𝑢₂ = 0. After collision 𝑣₁ = −𝑢₁ and 𝑣₂ = 0 that is, the small mass 𝑚1 rebounces with the same speed, whereas the large mass 𝑚₂ remains at rest. For example, a ball that is dropped from a height hits the floor and if the collision is perfectly elastic, it would re-bounce with the same speed and rise to the same height from which it was released. 

If 𝑚₁ is the mass of a ping pong ball and 𝑚₂ is the mass of a bowling ball. If ping pong ball collides with bowling ball, it will bounce back 

Case 3 

if 𝑚₁ = 𝑚₂ then 𝑣₁ = 𝑢₂ and 𝑣₂ = 𝑢₁ 

That is, in an elastic collision between two bodies of the same mass, the velocities of the bodies interchange after collision. For example if one body is initially at rest 𝑢₁ = 0 , then after collision, 𝑣₁ = 0 and 𝑣₂ = 𝑣₁. That is the moving body stops and the stationary body moves off with the velocity of the first body.