Imagine kicking a ball along the ground. It will travel a certain distance, depending on how hard you swing your leg, and will come to a stop. It will not carry on forever due to frictional forces. However, if you were deep in outer space and kicked the same ball, it would keep traveling at the velocity that you kicked it forever (at least until it collided with something or interacted with the gravitational field of a large mass). This is because all objects with mass have inertia, which means that they will not change their state of motion unless affected by an external force. They will also remain at rest if they are left alone. Carry on moving through this article with your inertia!
Inertia Definition
Inertia is the tendency of an object to resist a change in its state of motion.
Fig. 1. Both objects in rest and in motion have inertia.
To get an object moving or to bring a moving object to a halt, a net force needs to be acting on the object. A net force causes an acceleration. The forces on an object can be easily visualized using a free-body diagram, as shown below. The length of each arrow represents the magnitude of the force. Forces of the same magnitude pointing in opposite directions will cancel each other out, but if they are different then there will be a net force, which will cause acceleration or deceleration.
For example, if a car is moving along a road, it feels the following forces:
- Thrust from the engine \( F_E \)
- Rolling friction from the tires on the road \( F_F \)
- A drag force caused by air resistance \( F_D \)
- Gravity \( F_G \)
- A normal reaction force from the ground \( F_N \)
All of these forces can be considered to act at the center of mass of the car.
Fig. 2. The forces acting on a car can be represented by a free-body diagram.
The force required for a given acceleration is directly proportional to the inertia of the object. The inertia is dependent on the mass of the body. Smaller bodies with little inertia can be accelerated with a small force. On earth, most objects will eventually stop moving due to frictional forces. In the absence of friction, such as in the vacuum of space, objects will continue to move indefinitely due to their inertia.
A tablecloth can be pulled out from beneath a set dining table with a simple rapid pull. These dishes and cutlery will remain in their position due to their inertia.
Law of Inertia
Objects at rest will remain at rest and objects that are in motion will continue to move. A change of motion will occur only when a net force acts on an object. Wait! haven't we heard that somewhere before? Newton's first law of motion is also called the law of inertia.
A body at rest or moving at a constant speed in a straight line, will remain at rest or keep moving in a straight line at constant speed unless it is acted upon by an external unbalanced force.
Everything with mass has inertia, which is why we need to apply an external force to produce or stop motion. Mass is not to be confused with the weight of an object. Weight is the force produced when mass is present in a gravitational field such as that of the Earth. Remember the mass of a body is always constant but the weight will depend on the gravitational field it is in. For instance, objects in outer space that are very far away from anything else (so that they are not affected by any gravitational field) have a mass but do not have a weight.
Inertia Equation
As mentioned above, the inertia of an object is its tendency to resist a change in motion. When an object's motion is changing, it is said to be accelerating. Acceleration is caused by a net force and this was summed up by Newton in his second law, which can be stated as:
A net force acting on an object causes it to accelerate and its acceleration is proportional to, and in the same direction as, the net force.
Newton's law can also be expressed as the equation
$$\vec F=m\vec a,$$
where \( \vec F \) is the net force acting on the object measured in \( \mathrm N \), \( m \) is the mass of the object measured in \( \mathrm kg \) and \( \vec a \) is its acceleration in \( \mathrm m/\mathrm s^2 \). Notice that the force and the acceleration are vectors and both of these vectors point in the same direction.
Vector quantities have both a magnitude and a direction.
This equation says that the acceleration of an object is directly proportional to the net force acting on it, with the constant of proportionality being the mass. However, we can look at this equation in a different way. If we rearrange it to find the acceleration in terms of the force and mass as follows
$$\vec a=\frac {\vec F}{m}.$$
Suppose that we apply a given force to a range of objects with different masses. This equation says that the acceleration of an object is inversely proportional to its mass, so the objects with a larger mass will experience a smaller acceleration - they will have a greater resistance to change in motion! This brings us back to what we stated earlier - that the inertia of an object depends on its mass.
Usain Bolt is able to reach his top speed of \( 12\, \mathrm m/\mathrm s \) after \( 7\, \mathrm s \) when running from a standing start. He weighs \( 94\, \mathrm{kg} \) and his acceleration is approximately constant. What is the average accelerating force that he applies to the ground with his feet?
Fig. 3. Usain Bolt accelerates by pushing back against the floor with his feet
The equation for acceleration is given by
$$a=\frac{\Delta v}{\Delta t},$$
where \( \Delta v \) represents the change in velocity and \( \Delta t \) is the change in time. Using the values for the Bolt's maximum speed and the time taken to reach this speed, his acceleration can be found as
$$a=\frac{12\;\mathrm m/\mathrm s}{7\;\mathrm s}=1.7\;\mathrm m/\mathrm s^2.$$
To find the average force he exerts on the ground, we can use Newton's second law,
$$F=ma.$$
We are given his mass in the question so the average force is:
$$F=94\;\mathrm{kg}\times1.7\;\mathrm m/\mathrm s^2=160\;\mathrm{kg\,m}/\mathrm s^2=160\;\mathrm N$$
Types of Inertia
There are actually two types of inertia. The type we have been discussing so far is the inertia of linear motion (motion in a straight line). However, there is another type of inertia associated with objects in rotational motion.
Moment of Inertia
The moment of inertia of an object is its tendency to resist a change in its rotational motion about an axis of rotation.
It is called the "moment" of inertia because this inertia acts at a distance from the center of rotation. Linear inertia acts from the center of mass of the object. Just like how we discovered that an object with a greater mass will experience a smaller acceleration for a given force, an object with a greater moment of inertia will experience a smaller angular acceleration for a given torque.
Torque is a force that produces a rotation.
If an object can rotate about an axis, it will experience an angular acceleration about this axis if a torque is applied. A torque is due to a net force acting at a distance from the axis of rotation and is given by:
$$\tau=Fr,$$
where \( F \) is the force in \( \mathrm N \) and \( r \) is the perpendicular distance of the line of action of the force from the axis of rotation in \( \mathrm m \). Torque is measured in units of \( \mathrm{N\,m} \). Using Newton's second law for force, this equation becomes:
$$\tau=mar.$$
A torque causes an angular acceleration, so we need to have angular acceleration in the equation to find the relationship between them. Angular acceleration \( \alpha \) can be expressed in terms of an object's linear acceleration and its distance from the axis of rotation as:
$$\alpha=\frac ar,$$
which can be rearranged to:
$$a=\alpha r.$$
This can then be substituted into the above equation for torque to find the relationship between torque and angular acceleration as:
$$\tau=mr^2\alpha.$$
This is the rotational equivalent of Newton's second law. In Newton's second law, the constant of proportionality is the mass of the object \( m \), which describes the linear inertia of the object.
For the rotational motion equation, the constant of proportionality between torque and angular acceleration is \( mr^2 \), which is the equation for the moment of inertia of a point with mass \( m \) rotating about an axis at a distance \( r \). The moment of inertia \( I \) is measured in units of \( \mathrm{N\,m^2} \).
$$I=mr^2$$
Moment of inertia is measured in units of \( \mathrm{kg\,m}^2 \). If there are many point masses rotating about an axis, the moment of inertia is given by the sum of their moments of inertia:
$$I=\sum mr^2.$$
The moment of inertia of a rotating object depends on its mass distribution. The more mass concentrated further away from the axis of rotation, the greater the moment of inertia per unit mass. Some moments of inertia of various uniform shapes are given in the table below.
| Shape number | Shape | Moment of inertia |
| 1 | Point mass rotating about a radius r | \( I=mr^2 \) |
| 2 | Disk rotating about its center | \( I=\frac{mr^2}2 \) |
| 3 | Hollow cylinder rotating about its center | \( I=mr^2 \) |
| 4 | Solid sphere about its center | \( I=\frac25mr^2 \) |
| 5 | Solid cylinder about its center | \( I=\frac{mr^2}2 \) |
| 6 | Rod rotating about an axis perpendicular to itself passing through its center | \( I=\frac1{12}mr^2 \) |
| 7 | Rod rotating about an axis perpendicular to itself passing through one of its endpoints | \( I=\frac13mr^2 \) |
| 8 | Hollow sphere about its center | \( I=\frac23mr^2 \) |
Inertia Examples
Two blocks are pushed along a frictionless table with the same force \( F \). Block \( 1 \) has a mass of \( M \) and block \( 2 \) has a mass of \( 2M \). If the acceleration of block \( 1 \) is \( a_1 \), what is the acceleration of block \( 2 \), \( a_2 \), in terms of \( a_1 \)? Which block has the higher inertia?
For this question, we need to use Newton's second law equation,
$$F=ma.$$
We are asked to find acceleration, so the equation should be rearranged to get
$$a=\frac Fm,$$
so we can find \( a_1 \) as
$$a_1=\frac FM$$
and \( a_2 \) as
$$a_2=\frac F{2M}.$$
We need to find \( a_2 \) in terms of \( a_1 \). To do this, we can first express \( a_2 \) as
$$a_2=ka_1$$
in which \( k \) is a numerical constant. This expression can be rearranged to find \( k \) as
$$k=\frac{a_2}{a_1}=\frac{\displaystyle\frac F{2M}}{\displaystyle\frac FM}=\frac M{2M}=\frac12,$$
so we have found \( a_2 \) in terms of \( a_1 \):
$$a_2=\frac{a_1}{2}.$$
This means that the acceleration of block \( 2 \) is half of the acceleration of block \( 1 \), so it has a greater resistance to a change in motion and hence greater inertia. This is as expected, as block \( 2 \) has a greater mass.
A frisbee spins in the air as it is thrown. If a frisbee has a mass of \( 0.2\;\mathrm kg \) and a radius of \( 0.2\;\mathrm m \), what is its moment of inertia about its center? The frisbee can be modeled as a disk of uniform mass.
Fig. 4. A frisbee spins about an axis through its center when it is thrown
The moment of inertia of a disk is
$$I_d=\frac12mr^2,$$
where \( m \) is its mass and \( r \) is its radius. We are given both of these values in the question so we can calculate the moment of inertia as
$$I_d=\frac12\times0.2\;\mathrm{kg}\times{(0.2\;\mathrm m)}^2=0.04\;\mathrm{kg\,m}^2.$$
Inertia - Key takeaways
- Inertia is the tendency of an object to resist a change in its state of rest or motion.
- In the absence of friction or other external forces, objects will continue to move due to their inertia.
- An object with a larger mass has higher inertia.
- For a given force, the acceleration of an object is inversely proportional to its mass.
- The moment of inertia of an object is its tendency to resist a change in its rotational motion about an axis of rotation.
- Torque is a twisting force that produces a rotation.
- The formula for calculating the moment of inertia of a point mass rotating about an axis is given by \( I=mr^2 \).
- The moment of inertia of an object depends on its mass distribution about the axis of rotation.
References
- Fig. 1 - 'Usain Bolt Olympics Celebration' (https://upload.wikimedia.org/wikipedia/commons/7/7e/Usain_Bolt_Olympics_Celebration.jpg) by Richard Giles (https://www.flickr.com/people/35034356424@N01) is licensed by CC BY-SA 2.0 (https://creativecommons.org/licenses/by-sa/2.0)
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