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Course: AP®︎/College Physics 1 > Unit 7
Lesson 4: Rotational inertia and angular second lawRotational inertia and angular second law review
Overview of the key terms, equations, and skills related to rotational inertia, including how to analyze rotation inertia and how it relates to Newton's second law.
Key terms
Term (symbol) | Meaning | |
---|---|---|
Rotational inertia ( | Resistance to change in rotational velocity around an axis of rotation. Proportional to the mass and affected by the distribution of mass. Also called the moment of inertia. Scalar quantity with SI units of |
Equations
Equation | Symbols | Meaning in words |
---|---|---|
Angular acceleration is proportional to net torque and inversely proportional to rotational inertia. |
Analyzing rotational inertia
Rotational inertia depends both on an object’s mass and how the mass is distributed relative to the axis of rotation. Unlike other scenarios in physics where we simplify situations by pretending we have a point mass, the shape of an object determines its rotational inertia. We can’t just consider the mass to be concentrated at its center of mass.
When a mass moves further from the axis of rotation it becomes more difficult to change the rotational velocity of the system. For example, if we compare the rotational inertia for a hoop and a disc, both with the same mass and radius, the hoop will have a higher rotational inertia because the mass is distributed farther away from the axis of rotation.
If two objects have the same shape but different mass, the heavier one will have a larger moment of inertia.
How does rotational inertia relate to Newton’s second law?
Newton’s 2nd law relates force to acceleration. In the angular version of Newton’s 2nd law, torque takes the place of force and rotational inertia takes the place of mass. When the rotational inertia of an object is constant, the angular acceleration is proportional to torque.
For example, if we attach a rotating disc to a massless rope and then pull on the rope with constant force, we can see that the angular acceleration of the disc will increase as the force (and the torque) increases. A graph of the angular acceleration vs. torque would have a positive and constant slope because angular acceleration is directly proportional to torque . (See figure 2 below)
Common mistakes and misconceptions
- People sometimes forget that angular acceleration can be zero. If the torques on an object cancel out, the net torque is zero and the angular acceleration is also zero. For example, a beam that can rotate about its axis has two forces exerted on it and therefore two torques (see figure 3 below). Since the torques are in opposite directions, the net torque is zero and the beam will not rotate.
- Another common misconception is that the torques only sum to zero about the fulcrum. For an object in equilibrium, the torques sum to zero about any axis.
Learn more
For deeper explanations of rotational inertia, see our video on the rotational version of Newton's second law.
To check your understanding and work toward mastering these concepts, check out our exercises:
Want to join the conversation?
- For anyone out there also having trouble understanding why you must get farther to increase the acceleration.
Alpha = Torque / Moment of Inertia
Torque = Force * radius (radius of applied force)
Moment of Inertia = mass * radius^2 (radius of the object).
However, for a beginner like me it's very easy to think that r in Torque is the same as the r in Moment of Inertia, because of the simple mistake that both are written as r in the formula, and there is no clarification in the notes section.(19 votes)- i got a test tomorrow morning but THANK YOU-this got me messed up so bad because I keep getting confused when we change the radius of a hoop with the same mass, how does the torque increase if torque also equals force times r.(1 vote)
- given that the formula for the angular acceleration is:
alpha = torque/mr^2.
Under the assumption that the force is applied in a 90 degree angle it can be simplified to:
F*r/mr^2 which leads to: alpha = F/mr.
Why is it that in order to get the highest angular acceleration the force must applied at the largest distance from the pivot point when the acceleration is inversely proportional to the distance.
thanks in advance.(6 votes)- Your argument is okay when you deal with a mass attached to a massless rope, but when you are dealing with objects like discs, hoops or cylinders they have an specific moment of inertia that is given by their mass and their geometrical characteristics. I.e if we deal with the same disc its moment of inertia is a constant because its mass an its geometrical characteristics doesn't change independently the place we apply the force.
Moment of inertia of different objects:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/mic.png(3 votes)
- How are the units supposed to work out? How is a (kg*m^2)(rad/s^2) = n/m?(4 votes)
- so I took the practice: Angular acceleration and angular second law, and I kept failing. The correction was "Applying the force farther from the axis of rotation increases the angular acceleration, so we should decrease the distance instead." Meanwhile, in the video, David said that an increase in r, results in a decrease in angular acceleration. Why the different school of thoughts?(2 votes)
- No, an increase in r makes a larger angular acceleration because increasing r means that you are increasing the torque. The more torque you exert, the more angular acceleration you will have. Hope this helps!(3 votes)
- "Another common misconception is that the torques only sum to zero about the fulcrum. For an object in equilibrium, the torques sum to zero about *any axis.*" What does this mean? If the axis changes, doesn't the torques also changes since r changes in the equation torque=r Fnet?(3 votes)
- It is false that "if the net force on an object with fixed pivot is zero,then its net torque is also zero.
What is the reason?(1 vote)- The statement is true.
If the net force is zero, the object is at rest because there is no unbalanced force acting upon it. Torque is force times distance (or radius). If the force is zero, the torque is also zero because they are directly proportional to each other.(2 votes)
- if you have a bar fixed to the wall at a point, will the torque be greater if you push the bar at a point farther away from the wall (fulcrum)?(1 vote)
- Yes, torque = r*F*sin(theta), so increasing the radius will increase the torque.(1 vote)
- Can you please explain the following statement,Another common misconception is that the torques only sum to zero about the fulcrum. For an object in equilibrium, the torques sum to zero about any axis(1 vote)
- How angular acceleration increases if we increase the radius of location the of force? Will it not increase the moment of inertia?(1 vote)
- What exactly is the relationship between an object's rotational acceleration and mass?(1 vote)