Inertia

This page defines and describes inertia.

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 is in fact sometimes 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.

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.

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. This was 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 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. To begin, 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 stopped moving if conditions were ideal. This led to his idea of inertia.

Calculations with Inertia

Mass denoted by m is more accurately referred to as inertial mass. That is because the true definition of mass is a body's resistance to motion. Oftentimes mass is incorrectly thought of with regard to weight, which is the force of gravity acting upon a body. Inertial mass is present in a majority of kinematics equations, such as F=ma, and p=mv.

The formula for the standard moment of inertia, typically denoted as I, depends on what the particular statically determinate object is as well as its rotational axis.

Calculating the polar moment of inertia, typically denoted as J, depends on what the deformable body is and also depends on the rotational axis; the cross sectional area, distance from the center of mass, and projected deformation are also key components. J comes from the relationship between the total internal torque at a cross section and the stress distribution at a cross section.

Inertial Reference Frame

Reference frames are important for how velocity and acceleration are perceived. Imagine a person sitting in a car moving at a constant speed due north. To that persons left there is a truck moving at the same speed due north. To the persons right there is a pedestrian moving much slower but in the same direction. Think of the reference frame as the point of view of the person in the car. From their perspective the truck would be standing still and the pedestrian would be moving backwards. This is because in this reference frame the person in the car is being treated as a fixed point. All motion is perceived relative to that fixed point. The distance between the car and the pedestrian is increasing which is why the pedestrian appears to be moving away.

Examples:

If object A is moving at 1 mph and object B is moving behind object A it at 2 mph, how fast is object be moving relative to object A?

-In this example from the perspective of object A, object B is approaching object A at 1 mph

What if object A is moving 1 mph towards object B?

-In this example from the perspective of object A, object B is approaching object A at 3 mph

Important Concepts:

-The velocity of the reference frame needs to be taken into consideration when computing inertial reference frame problems.

It's important to note that this concept also applies to acceleration. If someone were to drop a ball from an accelerating car the ball would appear to have a force accelerating it in the reverse direction. This force would not actually exist. From the perspective on someone standing on the side of the road the ball would appear to move in the direction of the car (inheriting the velocity of the car) and downwards ( due to gravity).

Example: Person A is in a car accelerating North at 15 m/s^2. Describe the motion of a ball dropped from the car after 5 seconds.

-The ball would appear to accelerate down at 9.81m/s^2 due to gravity, and would move 75 m/s north.

Describe the motion of the ball in reference to the car.

-The ball would appear to accelerate down at 9.81m/s^2 due to gravity, and would accelerate 15m/s^2 south. -keep in mind the ball isn't truly undergoing that acceleration south. It just appears to from that reference frame.

Examples

Simple

A simple example of inertia is the classic trick of pulling a table cloth off of a table while the plates and cups stay in place. The tablecloth is acted upon by a force applied through the persons arm, while the only horizontal force acting on the plates and cups is friction. The person performing this trick must keep that in mind because a tablecloth with too high of a friction coefficient will act on the plates and cups and cause them to accelerate off of the table. It is clear from this example, though, that the plates and cups remain still unless acted upon by an outside force due to their inertia.

Meddling

Area Moment of InertiExample.jpg:

Difficult

Polar Moment of Inertia Example:

See also

Galileo Galilei

Conservation of Momentum

Newton's Laws and Linear Momentum

The Moments of Inertia

Acceleration

Velocity

References

[1] [2] [3] [4]