Center of Mass -5

VARIABLE MASS:

These are the systems in which mass does not remain constant with respect to time.

Basic Equation:

Derivation:

The two states of the system are shown at instants t and t + Dt. Initially whole mass M moves with . After Dt time, DM moves with relative velocity while rest (M-DM) moves with . Note that absolute velocity of DM is .

On the system

Now,

Note: Here is negative, so thrust force becomes +ve.

Illustration: A rocket of initial mass m₀ (including shell and fuel) is fired vertically at time t = 0. The fuel is consumed at a rate of ‘a = At’ where A = constant, t = time. The gases are exhaust at a constant relative speed of ‘m’ m/s with respect to rocket. In addition to gravity, there is resistance from air which can be assumed to be a constant force of . Find the differential equation velocity of rocket at the time t?

Solution:

{This is the mass at time t}

Let the velocity of rocket at time ‘t’ = V.

This is the differential equation governing the velocity of rocket with respect to time.

Illustration:

The chain has mass per unit length = M kg/m. What should be the value of F so as to pull the chain with constant velocity v?

Solution:

Let at the time t

Dumb Question:

<!--[if !supportLists]--> 1)      What is ?

Ans: If ‘dx’ length is also set into motion then mdx mass is set into motion.

<!--[if !supportLists]--> 2)      What is the system here?

Ans: The system can be the chain that is being set into motion. Solving with time, the system is increasing in length and hence mass!