Newton’s Second Law in Rotational Form
What happens when the torques on a system do not cancel.
Describe the conditions under which a system’s angular velocity changes.
Everything on this page serves that one sentence.
Topic 5.5 was about torques canceling. This one is about what happens when they do not. You already know the general answer, because you met it in Unit 2 with forces and masses, and the rotational version has the same three pieces in the same arrangement.
A net torque is the cause, an angular acceleration is the effect, and the rotational inertia determines how much effect you get for a given cause. Before you read on, try to write that sentence as an equation yourself. You have met all three symbols already.
1. A net torque changes the angular velocity of a system
When the torques on a system do not add to zero, the angular velocity changes. That is the first statement in full, and it matches exactly what a net force does to an ordinary velocity.
The change is in the direction of the net torque. When the net torque is in the same direction as the spin, the spin gets faster. When it is in the opposite direction, the spin slows down, and if that torque continues to act, the system stops and then turns the other way.
One thing does not matter here, and it is easy to miss. The angular velocity a system has does not tell you anything about the net torque on it, and a system turning quickly may have no net torque at all. Only the change in angular velocity is evidence of a torque.
Below is a wheel with a torque you set, starting at rest. Press Run and a graph of angular velocity against time is drawn as it turns, with the slope marked. Reverse the torque part way through and watch the slope of the graph change sign while the wheel is still turning in its original direction.
Test thisPress Run, let the wheel speed up for a few seconds, then press Reverse the torque without pausing.
Report what happenedAt the moment you reversed the torque, what happened to the slope of the graph, and what happened to the angular velocity? Explain why the slope can be negative for a while when the angular velocity is still positive.
Now without the simulationA wheel is turning counterclockwise and slowing down. The net torque on the wheel is
Topic questions
1 A system has a large angular velocity. It follows that
2 A net torque acts on a wheel in the same direction as its rotation. The wheel
2. Newton’s second law in rotational form
The three quantities are related in the way you would expect from Unit 2.
\(\alpha_{\text{sys}} = \frac{\sum \tau}{I_{\text{sys}}} = \frac{\tau_{\text{net}}}{I_{\text{sys}}}\)the counterpart of \(a = F_{\text{net}}/m\)
You read it the way you read the linear version. The angular acceleration is directly proportional to the net torque, so twice the torque gives you twice the angular acceleration. It is inversely proportional to the rotational inertia, so twice the rotational inertia gives you half the angular acceleration for the same torque.
Rotational inertia is in the denominator, which is where mass was before. That is the whole reason \(I\) was worth a topic of its own: it is the number that determines how much angular acceleration a given torque produces.
Worth remembering. Three equations now have the same shape, so keep them apart instead of only noticing that they match. A force goes with a mass and a linear acceleration. A torque goes with a rotational inertia and an angular acceleration. Putting an \(I\) where an \(m\) belongs gives you an answer with the wrong units, so the units are the first thing you should check.
Below are sliders for the net torque and for the rotational inertia, with the angular acceleration calculated as you move them and shown as a third bar. The plot shows you the angular acceleration against the rotational inertia at the torque you have set, so you can see the inverse relationship as a curve.
Test thisSet a torque and note the angular acceleration. Double the torque and read it again. Return the torque to its first value and then double the rotational inertia instead.
Report what happenedWhat did doubling the torque do, and what did doubling the rotational inertia do? Explain which of the two is in the numerator and which is in the denominator.
Now without the simulationA net torque of \(6.0\) N m acts on a system with rotational inertia \(3.0\) kg m\(^2\). Its angular acceleration is
Topic questions
3 The net torque on a system is doubled and nothing else is changed. Its angular acceleration
4 Two wheels receive the same net torque. Wheel A has three times the rotational inertia of wheel B. Comparing their angular accelerations,
3. The same torque on different systems
When you put the last two topics together, you get the result that makes rotational inertia worth knowing. Give three wheels of the same mass and the same radius the same torque, and they will not have the same angular acceleration, because they do not have the same rotational inertia.
The wheel with its mass pulled in toward the axis has the largest angular acceleration in each case, and you can calculate that factor before the race starts. None of this depends on how heavy the wheels are, since they are equally heavy.
This is also the answer to a question that comes up often in engineering. When you want something to change its spin quickly, you move mass inward. When you want it to keep a steady spin while small torques act on it, you move mass outward, which is why a flywheel is built as a heavy rim rather than a solid block.
Below are three wheels of equal mass and equal outer radius, with the mass arranged differently in each, and the same torque applied to all three when you press Run. There is a finish line at a fixed number of turns, so you can see which wheel arrives first, and the panel shows you each rotational inertia next to each angular acceleration.
Test thisPress Run and watch all three reach the finish line. Read the three rotational inertias and the three angular accelerations.
Report what happenedWhich wheel arrived first, and what was its rotational inertia compared with the others? Explain in one sentence why equal masses did not give equal angular accelerations.
Now without the simulationThree wheels of equal mass and equal radius are given the same torque. The one that reaches a given angular velocity first is the one with
Topic questions
5 A flywheel is designed to keep machinery turning at a steady rate despite small disturbances. It should be built with
6 Two wheels have the same mass and the same radius but different rotational inertias. Given the same torque, they differ in
4. Two analyses, done separately, connected by one equation
Many real systems move and turn at the same time, and the way to handle them is not to look for a single equation that covers both. It is to write the linear analysis for the part that moves and the rotational analysis for the part that turns, and then connect them.
A block hanging from a string wrapped around a pulley is the standard example. For the block you use Newton’s second law in its ordinary form, with its weight pulling down and the string tension pulling up. For the pulley you use the rotational form, with the tension in the string as the only source of torque about the axle.
The string is what connects the two analyses. As long as it does not slip or stretch, the linear acceleration of the string is the radius multiplied by the angular acceleration of the pulley, which is the same relationship you met in Topic 5.2. Those three statements together are enough to solve for every unknown.
\(mg – T = ma, \qquad TR = I\alpha, \qquad a = R\alpha\)three equations, three unknowns
Here is the result that surprises people. The tension is not the weight of the block. It cannot be, because the block is accelerating downward, and an upward pull equal to the weight would keep it at rest. The harder the pulley is to turn, the closer the tension gets to the full weight and the slower the block falls.
Below is a block hanging from a string wound around a pulley, with sliders for the block’s mass, the pulley’s rotational inertia and its radius. The two analyses are shown as two boxed panels, the block’s on one side and the pulley’s on the other, with the equation that connects them in between. Press Release to start the motion.
Test thisPress Release and read the tension. Compare it with the weight of the block, which the panel also shows you. Now increase the rotational inertia of the pulley and release it again.
Report what happenedIs the tension ever equal to the weight? Explain what happens to the tension and to the falling speed as the pulley becomes harder to turn.
Now without the simulationA block falls while unwinding a string from a pulley. The tension in the string is
Topic questions
7 A string wrapped around a pulley does not slip. The relationship between the linear acceleration of the string and the angular acceleration of the pulley is
8 In a block and pulley problem, the usual approach is to
That is Unit 5 on the site. The unit problem set and the practice quiz follow the same order as these six pages.