First thing that we must clearly understand that invariance of mass is a scientific reality only in restricted sense. We know that mass of a body depends on its speed and that the change is significant at high speed. However if we exclude the change in mass of a body at high speed, then “given” mass is essentially invariant in classical mechanics (where speed is relatively smaller with respect to speed of light).

** Mass invariance in classical mechanics **

Invariance of mass is a central theme in Newtonian mechanics. Even body systems like rocket propelled by escaping gas or leaking balloon are cases of redistribution of mass in body systems rather than variance of given mass per se. This mass invariance in classical mechanics enables us to take the mass out of the differential equation given by Newton’s second law :

This is important from the perspective that external force can be expressed in terms of acceleration :

A situation where measurement of force is required, therefore, assumes that mass is invariant. We proceed to measure force either in terms of acceleration or change in momentum in equivalent manner.

** Force as product of mass and acceleration **

##### Example 1

Problem : A bullet of 10 gram enters into a wooden target with a speed of 200 m/s and comes to stop with constant deceleration. If linear penetration into the wood is 10 cm. find the force on the bullet.

A bullet hits wooden target |
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Solution : The motion of the bullet is resisted by wood, which applies a force in a direction opposite to that of the bullet. We can find out the force on the bullet, if we know the deceleration during the motion. Now, as deceleration is constant, we can apply equation of motion. Here,

u = 500 m/s ; v = 0; x = 10/100 = 0.1 m.

Force on the bullet,

F = ma = - 0.01 x 200000 = -2000 N

Note that bullet, in turn, applies 2000 N of force on the wooden target. This explains why bullet hits are so fatal for animals.

** Force as a time rate of change in linear momentum **

##### Example 2

Problem : A ball of mass “m” with a speed “v” hits a hard surface as shown in the figure. The ball rebounds with the same speed and at the same angle, θ, with vertical as before. Find average force acting on the ball.

A ball strikes the hard surface |
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Solution : We can determine force either (i) by measuring acceleration or (ii) by measuring change in linear momentum during the motion. Here, we take the second approach.

The linear momentum (remember that it is a vector) before hitting surface is :

Components of linear momentum before hitting surface |
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The linear momentum (remember that it is a vector) after hitting surface is :

Components of linear momentum after hitting surface |
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The change in linear during contact with surface, Δp,

The average force is :

Note that force is acting in vertical upward direction.

** Mass variance in classical mechanics **

However, we come across situation even in Newtonian mechanics, in which we find that mass of a body under study is actually changing. This change results as there is exchange of mass between the body system and its surrounding. Th motion of rocket launched in space or a leaking balloon are examples of changing mass system. But, we must understand that this is not the change of “given” mass; rather mass is simply distributed from one body system (say balloon)into two or more body systems (balloon and its surrounding).

Leaking balloon |
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Now, does this variance of mass have any bearing on the form of Newton’s second law or about the meaning of force as acceleration of unit mass? This is the question that we need to answer. Now, let us consider the situation in terms of Newton’s second law (Here we consider one dimensional case only to keep the discussion focused) :

Apparently, this mass variant form of Newton’s second law appears to destroy the well founded meaning of force as the acceleration of unit mass. As a matter of fact, it is not so.

Let us recall the relation, F = ma, which essentially underlines relation between force (cause) and acceleration (effect). Now look closely at the additional term resulting from change in mass in the context of a real time case like that of a rocket. Here, the body i.e. rocket is loosing mass with the escaping gas and hence dm < 0 i.e. change in mass is negative.

The rocket is put into motion with expulsion of high speed gas. The escaping gas provides the thrust (force) to accelerate the rocket. What it means that the escaping gas mass is yet another body system which is applying force on the rocket and thereby constitutes external force to the rocket.

The additional term, therefore, is not an “effect” term like “ma”, but a term representing “external force”. Rewriting the equation to segregate cause and effect terms on either side of the equality, we have :

Thus, we see that a varying mass does not change the fundamental aspect of Newton’s second law of motion. It only modifies the external force on the body. The net external force is still equal to the product of mass and acceleration.

#### Example 3

Problem : Air from a leaking balloon of mass 15 gm gushes out at a constant speed of 10 m/s. The balloon shrinks completely in 10 seconds and reduces to a mass of 5 gm. Find the average force on the balloon.

Solution : There is no external force on balloon initially. However, gushing air constitutes an exchange of mass between balloon and its surrounding. This generates an external force on the balloon given by :

Leaking balloon |
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We summarize the discussion held, in the context of classical mechanics, so far as :

- Given mass is invariant.
- The variance of the mass of body system is actually a redistribution of mass from one body system to more than one body systems.
- Irrespective of the nature of mass (varying or constant) of a body system, external force is equal to time rate change of momentum. As linear momentum is the product of mass and velocity, the linear momentum can change because of change in either of the two quantities in any combination.
- If there is redistribution of mass, then it results into an external force to the original body system, modifying the external force on the body.
- Irrespective of the nature of mass (varying or constant) of a body system, the net external force is equal to the product of mass and acceleration.
- Irrespective of the nature of mass (varying or constant) of a body system, momentum and acceleration forms of Newton’s second law are equivalent and consistent to each other.

** Mass variance in relativistic mechanics **

The detailed discussion of this topic is not part of the course. For information sake, we shall only outline the features of force law in relativistic mechanics. Here, momentum and acceleration forms are not equivalent. As a matter of fact, the momentum form is valid even in relativistic mechanics i.e.

However, the acceleration form is not valid. For relativistic mechanics, this form is written, in terms of rest mass,

Yet another important aspect of force here is that force and acceleration vectors need not be parallel or in the same direction.

We summarize the discussion held, in the context of relativistic mechanics, so far as :

- Given mass is not invariant.
- The momentum and acceleration forms of Newton’s second law are not equivalent.
- Momentum form of Newton’s second law is valid.
- Acceleration form of Newton’s second law is not valid.
- Force and acceleration need not be in the same direction.