Conservation of Momentum Circular Motion Projectile Motion

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Circular motion

Velocity is a vector quantity. Its magnitude is called speed and also it has a direction. A body of mass m is moving in a circular path of radius r. There are two possibilities:

First possibility

If the speed is constant, the motion is called uniform circular motion. Speed is constant but the direction is changing continuously, so the motion is accelerated. And the acceleration is along the radius towards the center.

This acceleration is called centripetal acceleration or even radial acceleration and is given by the relation:

a = v² / r

Where v is the speed. The direction of velocity is along the tangent to the circular path. Thus it changes as the body is at different positions at different times.

Centripetal force = Mass x Centripetal Acceleration

The relation between linear velocity v and angular velocity ω is v = ωr where ω = 2πf

f is the frequency of revolution (units rev/s) ω has the units rad/s

Since cetripetal force is always perpendicular to the velocity, it can not change the magnitude of velocity (speed) or it can not change its speed. It can only change its direction, thus making it move in circular path with constant speed.

Other relations for F:

Since v = ωr

Also

Second possibility

To change the speed of the body in circular motion, a tangential force has to be applied. If it acts along the direction of velocity, it will increase its speed, On the other hand if it acts opposite to the direction of velocity it will slow it down. Now the motion is called non-uniform circular motion

Net acceleration

In non-uniform circular motion, speed as well as direction changes continuously.

Center of mass

Consider that there are a number of particles forming a system, such that, when an external force is applied, the complete system moves as on. Now, if we are able to replace all these particles by just one particle. And, this new particle is placed in such a manner and has such a mass, that when acted on by an external force, it moves exactly in the same manner as the system it replaced. Then, we have found the center of mass of the system. (This sounds more complex than it is. So don't worry! read on)

Center of mass of a two particle system

Let us now consider two particles. Let them have masses m₁ and m₂. Let r₁(t) and r₂(t) be the position vectors of the two particles at the instant of time t. (Note: On a web page it is difficult to use the usual vector notation. So I will use bold fonts to depict vectors). The position vector R_c.m.(t), giving the center of mass of this system is given by the formula.

R_c.m.(t) = {m₁r₁(t) + m₂r₂(t)} / (m₁ + m₂)

from the above it is obvious that for a two particle system the center of mass always lies between the two particles. It also lies on the line joining the two particles.

Center of mass of a large number of particles

R_c.m.(t) = {m₁r₁(t) + m₂r₂(t) + ----- m_n r_n(t)} / (m₁ + m₂+ ------ m_n)

Center of mass of a rigid body

The position of the center of mass of a rigid body is a fixed point. This point may or may not lie in the body. Where is lies depends upon the shape of the body. For example of we take the case of a ring. The center of mass will lie at its geometrical center. Obviously this point does not lie in the ring.