Newton's Second Law in Angular Form

Torque and Force

Torque is the rotational analog of force because it is the measure of the force required for an object to start rotating about an axis. As in linear motion where a force causes an object to accelerate, torque causes the angular acceleration of an object.

Torque can be defined by three formulas.

1. Cross Product Formula,
2. Magnitude Formula,
3. Newtons Second Law Formula.

Cross-Product Formula

The cross-product definition of torque is expressed by the equation

$$\vec{\tau}=\vec{r} \times \vec{F}$$

where \( \vec{r} \) is the lever arm measured in \( \mathrm{m} \) and \( \vec{F} \) is the force applied measured in \( \mathrm{N}. \)

It is important to recognize that cross product is another term for vector product indicating that both \( r \) and \( F \) are vector quantities.

Magnitude Formula

As a result of the cross-product, the magnitude definition of torque is expressed by the equation

$$\tau=rF\sin\theta$$

where \( r \) is the lever arm, \( F \) is the applied force, and \( \theta \) is the angle between the lever arm and the applied force. The variables, \( r \) and \( F \), no longer represent vectors but rather correspond to the magnitude of each vector.

Angular Acceleration and Linear Acceleration

Angular acceleration is the rotational equivalent of linear acceleration.

This relationship is expressed by the formula \( a=\alpha{r} \) which can be rewritten as \( \alpha=\frac{a}{r} \) where \( r \) represents radius.

The mathematical formula corresponding to this definition is

$$\alpha=\frac{\Delta{\omega}}{\Delta{t}}$$ where \( \omega \) is angular velocity and \( t \) is time.

Moment of Inertia and Mass

The rotational analog of mass is inertia. An object's moment of inertia describes how mass is distributed relative to the axis of rotation. Meanwhile, mass itself describes the amount of matter within an object.

Formulas regarding an object's moment of inertia will vary depending on the shape of the object. For example, the moment of inertia of a point mass a distance \(r\) from the axis of rotation is given by \( I=mr^2. \) This formula is important because it is the basis for all other inertial formulas since objects can be constructed from sets of point masses.

Formula of Newton's Second Law for Rotation

The equation for Newton's second law in angular form is expressed as

$$\tau =I\alpha$$

where \( \tau \) is the total net torque measured in \( \mathrm{N\,m} \), \( I \) is the moment of inertia measured in \( \mathrm{{kg}\,{m^2}} \) and \( \alpha \) is angular acceleration measured in \( \mathrm{\frac{rad}{s^2}}. \)

Meaning of Newton's Second Law in Angular Form

Newton's second law in angular form states that angular acceleration is proportional to torque and inversely proportional to the moment of inertia. This relationship can be seen if the equation is rewritten in terms of angular acceleration as \( \alpha= \frac{\tau}{I} \). This means that a greater torque implies a greater angular acceleration, and a greater moment of inertia implies a smaller angular acceleration. The latter is what happens with your shopping cart!

Proof of Newton's Second Law in Angular Form

To prove that Newton's second law can be written in angular form, let us consider this example. An object with mass \( m\) is rotating about an axis and a force is exerted at a distance \( r \) from the axis. We know that the object is constrained to rotate in a circular motion as a result of the fixed axis and we assume that the force is tangent to the circle. Therefore, using this information in conjunction with the relationship between linear and angular acceleration, \( a=\alpha{r} \), enables the equation to be rewritten as follows:

\[\begin{align*}F&=ma,\\ rF&=rma,\\ rF&=rm(\alpha r),\\ rF&=(mr^2)\alpha,\\ \tau&=I\alpha.\end{align*}\]

Thus, we can derive the angular form of Newton's second equation from the linear form.

Common Applications of Newton's Second Law

A common application of Newton’s second law in angular form is a merry-go-round.

Figure 2: A merry-go-round as an example of Newton's second law in angular form.

If you have ever taken a child to the park, you have demonstrated Newton's second law in angular form by using the merry-go-round. Rotating an empty merry-go-round is much easier than rotating one with children on it. Why? Well, the answer is Newton's second law in angular form. When we spin the merry-go-round, we grab its edge and begin to apply a perpendicular force to the radius of the merry-go-round. This allows us to apply the maximum amount of torque necessary for the merry-go-round to accelerate. However, as soon as more mass is added, i.e., children climb onto it, the merry-go-round accelerates less easily because the additional mass results in a larger moment of inertia. If we wanted to increase the angular acceleration, we would have to apply more torque (and thus more force if we keep grabbing the edge).

Torque Examples with Newton's Second Law in Angular Form

To solve problems regarding Newton's second law in angular form, one can use the equation for torque and apply it to different problems. As we have defined torque and discussed its relation to rotational motion as well as multiple variables, let us work through some examples to gain familiarity with the equations.

Note that before solving a problem, we must always remember these simple steps.

1. Read the problem and identify all variables given within the problem.
2. Determine what the problem is asking and what formulas are needed.
3. Apply the necessary formulas and solve the problem.
4. Draw a picture if necessary to provide a visual aid.

Examples

Let us apply our newfound knowledge of Newton's second law in angular form to some examples.

Now, let's complete a similar example where we solve for the moment of inertia rather than torque.

Finally, let's end with a slightly more complicated example using the moment of inertia formula corresponding to a solid sphere.