What is Inertia?

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You may think of inertia as a mysterious force keeping you from doing something you have to do, like your homework, but that isn't what physicists mean by the word. In physics, inertia is the tendency of an object to remain at rest or in a state of uniform motion. This tendency is dependent on mass, but it isn't exactly the same thing. You can measure an object's inertia by applying a force to change its motion. Inertia is the tendency of the object to resist the applied force. Newton’s laws of motion are crucial in describing inertia as a property of matter, and it is important when considering everything from collisions and free fall to the impacts of Newton’s Third Law and the effect of an equal and opposite reaction.

The Concept of Inertia Comes From Newton's First Law of Motion

Because they seem so common sense today, it's hard to appreciate how revolutionary Newton's three Laws of Motion were to the scientific community of the time. Before Isaac Newton and Galileo, scientists had held a 2,000-year-old belief that objects had a natural tendency to come to rest if left alone. Galileo addressed this belief with an experiment involving inclined planes that faced each other. He concluded that a ball cycling up and down these planes would continue to rise to the same height forever if friction were not a factor. Newton used this result to formulate his First Law – also known as the law of inertia – which states:

Every object continues in its state of rest or motion in a straight line unless acted upon by an external force.

Physicists consider this statement the formal definition of inertia. It is a crucial metric when describing the net force and modeling kinematics for specific objects.

Inertia Varies With Mass

According to Newton's Second Law, the force (F) required to change the state of motion of an object is the product of the object's mass (m) and the acceleration produced by the force (a):

F = ma

To understand how mass is related to inertia, consider a constant force Fc acting on two different bodies. The first body has mass m1 and the second body has mass m2.

When acting on m1, Fc produces an acceleration a1:

F_c = m_1 a _1

When acting on m2, it produces an acceleration a2:

F_c = m_2 a_2

Since Fc is constant and doesn't change, the following is true:

m_1 a_1 = m_2 a_2

and

\frac{m_1}{m_2} = \frac{a_2}{a_1}

If m1 is bigger than m2, then you know a2 will be bigger than a1 to make both equal Fc, and vice versa.

In other words, the mass of the object is a measure of its tendency to resist the force and continue in the same state of motion. Although mass and inertia don't mean exactly the same thing, inertia is usually measured in units of mass. In the SI system, its units are grams and kilograms, and in the imperial system, the units are slugs. Scientists usually don't discuss inertia in motion problems. They usually discuss mass.

Moment of Inertia

A rotating body also has a tendency to resist forces, but because it's composed of a collection of particles that are at various distances from the center of rotation, scientists talk about its moment of inertia rather than its inertia (sometimes also called rotational inertia). The inertia of a body in linear motion can be equated to its mass, but calculating the moment of inertia of a rotating body is more complicated because it depends on the shape of the body and its circular motion. The generalized expression for the moment of inertia (I) or a rotating body of mass m and radius r is:

I = k m r^2

where k is a constant that depends on the shape of the body. The units of moment of inertia are (mass) • (axis-to-rotation-mass distance)2. Angular momentum and the moment of inertia are related concepts that help to describe the rotational motion of moving objects. We can also describe rotational kinetic energy as proportional to rotational inertia.

Inertia and Relativity

When Albert Einstein developed his theories of special and general relativity, he differentiated between inertial and non-inertial reference frames. Inertial reference frames hold a constant speed, while non-inertial reference frames account for acceleration and gravitation.

Special relativity uses these inertial reference frames to describe the effects of relativistic speeds that are close to the speed of light.

General relativity gets much more complex as it tries to include outside forces in the form of acceleration and gravity.

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