Inertia

Edited by MaKenna Kelly Spring 2021

This page defines and describes inertia.

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The Main Idea

Inertia is the tendency of matter to resist change in Velocity. It is an inherent property of matter; the inertia of an object is directly proportional to its Mass and can in fact be used to define mass. Newton's First Law of Motion states that the velocity of an object does not change unless there is an unbalanced force acting on it. This is a consequence of the object's inertia. When a net external force acts on an object, the object will accelerate, meaning its velocity will change over time. For a given force, the rate of change of velocity is inversely proportional to the mass of an object; more massive objects have more inertia and therefore experience slower changes in velocity for a given force.

"Inertia" is not to be confused with "moment of inertia", a related but different topic. The moment of inertia of an object is the tendency of an object to resist change in angular velocity. It is the rotational analogue for inertia. For more information about moments of inertia, see The Moments of Inertia.

Inertia is responsible fictitious forces, which are forces that appear to act on objects in accelerating frames of reference.

A Mathematical Model

According to Newton's Second Law: the Momentum Principle, [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math]. The more massive an object is, the less its velocity needs to change in order to achieve the same change in momentum in a given time interval.

The other form of Newton's Second Law is [math]\displaystyle{ \vec{F}_{net} = m \vec{a} }[/math]. Solving for acceleration yields [math]\displaystyle{ \vec{a} = \frac{\vec{F}_{net}}{m} }[/math]. This shows the inverse relationship between mass and acceleration for a given net force.

A Computational Model

Here is a link to a computational model that can aid in better understanding inertia. [1] The situation shown is very similar to that described in the "Industry Applications" portion of this page.

Here are snapshots from a similar model:

Inertial Reference Frames

In physics, properties such as objects' positions, velocities, and accelerations depend on the Frame of Reference used. Different reference frames can have different origins, and can even move and accelerate with respect to one another. According to Einstein's idea of relativity, one cannot tell the position or velocity of one's reference frame by performing experiments. For example, a person standing in an elevator with no windows cannot tell if it is moving up or down by, say, dropping a ball from shoulder height and measuring how long it takes to hit the ground. The answer will be the same regardless of the velocity of the elevator, in part because when the ball is released from shoulder height, it is travelling at the same velocity as the elevator. However, one can tell the acceleration of one's reference frame by performing experiments. For example, if a person standing in an elevator with no windows were to drop a ball from shoulder height, if the elevator were accelerating upwards, it would take less time for the ball to hit the ground than if the elevator were moving at a constant velocity. This is because the ground accelerates towards the ball. From the point of view of the person in the elevator, the ball would appear to violate Newton's second law; it would appear to accelerate towards the ground faster than accounted for by the force of gravity. This is because Newton's second law was written for inertial reference frames. An inertial reference frame is a reference frame travelling at a constant velocity. The physics in this course is true only in inertial reference frames, and problems in this course should be solved using inertial reference frames. The surface of the earth is usually considered an inertial reference frame; its rotations about its axis and revolutions around the sun are technically forms of acceleration, but both are so small that they have negligible effects on most aspects of motion. The definition of an inertial reference frame is one in which matter exhibits inertia; in accelerating reference frames, objects may accelerate despite the absence of external forces acting on them.

Examples

Simple

Find the mass of an object whose acceleration is 4 m/s^2 and whose translational inertia is 20 kg.

[math]\displaystyle{ T = ma }[/math]

Where T is translational inertia, m is mass, and a is acceleration.

[math]\displaystyle{ m = \frac{T}{a} }[/math]

[math]\displaystyle{ m = \frac{20 kg}{4 m/s^2} }[/math]

[math]\displaystyle{ m = 5 kg }[/math]

Middling

A motor capable of producing a constant torque of 100 Nm and a maximum rotation speed of 150 rad/s is connected to a flywheel with rotational inertia 0.1 kgm^2

a) What angular acceleration will the flywheel experience as the motor is switched on?

[math]\displaystyle{ \tau = I\alpha }[/math]

Where [math]\displaystyle{ \tau }[/math] is torque, I is rotational inertia, and [math]\displaystyle{ \alpha }[/math] is angular acceleration.

[math]\displaystyle{ \alpha = \frac{\tau}{I} }[/math]

[math]\displaystyle{ \alpha = \frac{100 Nm}{0.1 kgm^2} }[/math]

[math]\displaystyle{ \alpha = 1000 rad/s^2 }[/math]

b) How long will the flywheel take to reach a steady speed if starting from rest?

Using rotational kinematics,

[math]\displaystyle{ \omega = \omega_0 + \alpha \times t }[/math]

Since we know the maximum rotational velocity of the motor, we can solve to find the time taken to accelerate up to that rotational velocity.

[math]\displaystyle{ t = \frac{{\omega}_{max}}{\alpha} }[/math]

[math]\displaystyle{ t = \frac{150 rad/s}{1000 rad/s^2} }[/math]

[math]\displaystyle{ t = 0.15 s }[/math]

Difficult

What is the rotational inertia of the object shown above?

[math]\displaystyle{ I = m_1r_1^2 + m_2r_2^2 + ... = \sum m_ir_i^2 }[/math]

Where I is rotational inertia, m is mass, and r is distance from the axis of rotation.

[math]\displaystyle{ I = (1 kg \times 1^2 m^2) + (1 kg \times 1.5^2 m^2) + (1 kg \times 0.75^2 m^2) + (2 kg \times 0.75^2 m^2) }[/math]

[math]\displaystyle{ I = 4.9375 kgm^2 }[/math]

Connectedness

Interesting Applications

Scenario: Tablecloth Party Trick

A classic demonstration of inertia is a party trick in which a tablecloth is yanked out from underneath an assortment of dinnerware, which barely moves and remains on the table. The tablecloth accelerates because a strong external force- a person's arm- acts on it, but the only force acting on the dinnerware is kinetic friction with the sliding tablecloth. This force is significantly weaker, and if the tablecloth is pulled quickly enough, does not have enough time to impart a significant impulse on the dinnerware. This trick demonstrates the inertia of the dinnerware, which has an initial velocity of 0 and resists change in velocity.

Scenario: Turning car

You have probably experienced a situation in which you were driving or riding in a car when the driver takes a sharp turn. As a result, you were pressed against the side of the car to the outside of the turn. This is the result of your inertia; when the car changed direction, your body's natural tendency to continue moving in a straight line caused it to collide with the side of the car (or your seatbelt), which then applied enough normal force to cause your body to turn along with the car.

Connection to Mathematics

Inertia is studied using Newton's Second Law, which states that [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math]. The [math]\displaystyle{ \frac{d\vec{p}}{dt} }[/math] in this equation stands for the derivative of momentum.

Derivatives are core concept of a branch of Mathematics called Calculus, which studies constant motion. In this way, understanding inertia gives one a deeper understanding of Calculus, which is an integral member of the Mathematics family.

Industry Application

Using a seat belt is one of the best methods for preserving one's life in the event of a car crash. For this reason, they are a big deal in the vehicular safety industry.

Passengers in a moving car have the same motion as the car they are in. When that car stops abruptly, as cars often do in the event of a car crash, its passengers do not stop. Instead, they continue to have the same motion that the car did before it stopped because of inertia. This motion can often propel passengers of out the car, which increases the likelihood of death.

Seat belts are designed to fight against inertia and hold passengers in their seats, often saving their lives in the process.

History

Before Newton's Laws of Motion came to prominence, models of motion were based on of the observation that objects on Earth always ended up in a resting state regardless of their mass and initial velocity. Today we know this belief to be the result of Friction. Since friction is present for all macroscopic motion on earth, it was difficult for academics of the time to imagine that motion could exist without it, so it was not separated from the general motion of objects. However, this posed a problem: the perpetual motion of planets and other celestial bodies. Galileo Galilei (1564-1642) was the first to propose that perpetual motion was actually the natural state of objects, and that forces such as friction were necessary to bring them to rest or otherwise change their velocities.

Galileo performed an experiment with two ramps and a bronze ball. The two ramps were set up at the same angle of incline, facing each other. Galileo observed that if a ball was released on one of the ramps from a certain height, it would roll down that ramp and up the other and reach that same height. He then experimented with altering the angle of the second ramp. He observed that even when the second ramp was less steep than the first, the ball would reach the same height it was dropped from. (Today, this is known to be the result of conservation of energy.) Galileo reasoned that if the second ramp were removed entirely, and the ball rolled down the first ramp and onto a flat surface, it would never be able to reach the height it was dropped from, and would therefore never stop moving if conditions were ideal. This led to his idea of inertia.

References