Mass

Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. It is a central concept of classical mechanics and related subjects.

Inertial mass

To understand what the inertial mass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Later on, we will see how our classical definition of mass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics. However, the implications of special relativity will not change the meaning of "mass" in any essential way.

Related Topics:
Classical mechanics - Newton's Laws of Motion - Special relativity

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According to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion

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: F = rac{d}{dt} (mv)

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where F is the force acting on the body and v is its velocity. For the moment, we will put aside the question of what "force acting on the body" actually means.

Related Topics:
Force - Velocity

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Now, suppose that the mass of the body in question is a constant. This assumption, known as the conservation of mass, rests on the ideas that (i) mass is a measure of the amount of matter contained in a body, and (ii) matter can never be created or destroyed, only split up or recombined. These are very reasonable assumptions for everyday objects, though, as we will see, the situation gets more complicated when we take special relativity into account. Another point to note is that, even in classical mechanics, it is sometimes useful to treat the mass of an object as changing with time. For example, the mass of a rocket decreases as the rocket fires. However, this is an approximation, based on ignoring pieces of matter which enter or leave the system. In the case of the rocket, these pieces correspond to the ejected propellent; if we were to measure the total mass of the rocket and its propellent, we would find that it is conserved.

Related Topics:
Conservation of mass - Rocket

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When the mass of a body is constant, Newton's second law becomes

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: F = m rac{dv}{dt} = m a

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where a denotes the acceleration of the body.

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This equation illustrates how mass relates to the inertia of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force.

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However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of Newton's third law, which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects A and B, with constant inertial masses mA and mB. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on A by B, which we denote FAB, and the force exerted on B by A, which we denote FBA. As we have seen, Newton's second law states that

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:F_{AB} = m_A a_A , and F_{BA} = m_B a_B ,

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where aA and aB are the accelerations of A and B respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that

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:F_{AB} = - F_{BA} ,

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Substituting this into the previous equations, we obtain

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:m_A = - rac{a_B}{a_A} , m_B

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Note that our requirement that aA be non-zero ensures that the fraction is well-defined.

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This is, in principle, how we would measure the inertial mass of an object. We choose a "reference" object and define its mass mB as (say) 1 kilogram. Then we can measure the mass of every other object in the universe by colliding it with the reference object and measuring the accelerations.

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