7-Torque and Rotational Dynamics

Resources

Slideshows: Current Slideshow: Unit 5: Torque and Rotational Motion Notes
Make sure to open the presenter notes in the slideshow for important notes and answers.
Archived slideshows for additional practice scenarios: Rotational Kinematics notes, Rotational Dynamics (Torque, Inertia and acceleration)

Textbook: Chapters 7 & 8 in Mastering Physics (get online code for registration on about page of google classroom)

What is in this unit?

This unit builds models similar to unit 1 and 2 (forces and motion) that are adapted to describe and predict changes in rotational motion and the effect of forces on it.

Topic 5.1 Rotational Kinematics

Rigid System: a system that holds it shape, so different points move in different directions during rotation

Angular Displacement: the angle, in radians, through which a point on a rigid system rotates about a specified axis. [vector measured in radians]
Δθ = θ - θ0 **this definition is not on the formula chart

+ and - directions for rotation are assigned to either clockwise or counterclockwise directions

Average Angular Velocity: average angular velocity, ω, is the rate of change in angular position. [vector measured in radians/second]
ωavg = Δθ/Δt **this definition is not on the formula chart

Average Angular Acceleration: average angular acceleration, α, is the rate of change in angular velocity. [vector measured in radians/second squared]
αavg = Δω/Δt **this definition is not on the formula chart

Angular motion quantities are related the same way linear motion quantities are through kinematics

For constant acceleration rotation kinematics equations (see image of formula chart that follows) can describe, relate, and calculate expected values for rotational quantities.
Graphs of rotational kinematics quantities also relate in the same way as linear quantities.
- Slope of θ as a function of time is ω
- Slope of ω as a function of time is α
- Area under ω as a function of time is Δθ
- Area under α as a function of time is Δω

Topic 5.2 Connecting Linear and Rotational Motion

For a rigid system all points within that system have the same angular velocity and angular acceleration.

For any point at a distance r from a fixed axis, r can be used to convert between linear and angular quantities.

For s, linear distance by a point during rotation, Δs = rΔθ **this definition is not on the formula chart
For linear (tangential) velocity of a point v = rω
For linear (tangential) acceleration of a point aT = rα

Topic 5.3 Torque

Topic 5.4 Rotational Inertia

Topic 5.5 Rotational Equilibrium and Newton's 1st Law in Rotational Form

Topic 5.6 Newton's 2nd Law in Rotational Form

In addition to the equations above (on the formula chart), you should memorize or be able to recreate the following:

use r to convert between angular and linear quantities (v = ωr, a = ⍺r, d=𝜃r)
Replace angular quantities in linear equations -- like v2 = v02 + 2a(x-x0).
Create simple definitions for average angular quantities, like average angular acceleration = change in angular velocity divided by time, as we did with linear quantities
Know that I = mr2 for point objects at the edge of a circle or hoops, I = 1/2 mr2 for disks or cylinders, and is even less (2/5mr2) for spheres.
Show that the angular momentum of a linearly traveling object can be found by L=mvr (or in other words, L=pr)

Viral Physics videos

This is an amazing build project that actually uses what we talk about with torque, rotational energy, rotational inertia, and some really cool variations to create a near perfect transfer of energy.

11.8k Likes, 386 Comments - Dimitry Politov (@dimitrypolitov) on Instagram: “Two steps starfish run at the Bondi Beach in Sydney! First time doing it outside) Wearing…”www.instagram.com/p/BwIQgDboxQ0/?utm_source=ig_web_button_share_sheet

This dude has come insane core strength to hold this pose. How would you analyze the forces/torques in this equilibrium situation?

Common Misconceptions

- Misconception: Any force acting on an object will produce a torque.
  - Principle: Force components perpendicular to the radius (or forces with moment arms) cause torque
  - Reasoning: Torque is a measurement of a tendency to change rotation and in order to do that a force must in a way where the line of action of the force does not pass through the center of mass of a free moving object or the pivot point of a fixed object. If it does pass through these points it will affect the whole object equally and not cause rotation, where if it acts to one side or other of the pivot or center, it will cause that side to move along with the force while the other side pivots around the other way.

- Misconception: Objects moving in a straight line can not have angular momentum.
  - Principle: If an object can give angular momentum to a second object, it must have angular momentum to begin with.
  - Reasoning: Many systems/objects can be evaluated multiple ways. Straight line motion, if considered from a point of rotation that matters for the system, can can be described in terms of how many radians/s it moves as opposed to m/s. While the angular velocity will be increasing as it gets closer to the pivot, the r value will be decreasing and as a result the overall angular momentum of the object a remains constant L=mvr (as shown in the diagram where p=mv and D is the shortest R of the path. This is only meaningful if the angular momentum is going to be transferred to a second object; otherwise it is not intuitive.

- Misconception: Torque is the same as force and is in same direction.
  - Principle: Torque is a tendency to rotate caused by force with a direction that is 90 degrees to the force and moment arm as determined by the right hand rule.
  - Reasoning: The direction of torque is not terribly important until you talk about changing angular momentum, in which case we see weird stuff like gyroscopic precession (google it and be amazed). This behavior is explained most directly by assigning the direction for angular momentum and torque according to the right hand rule.

- Misconception: The direction of angular momentum is in direction of linear momentum.
  - Principle: The direction of angular momentum is determined by the right hand rule.
  - Reasoning: Use the same diagram as above for either angular speed or angular momentum, but alter it by using your fingers to represent the movement ov the rotating object, as opposed to the force acting on it. It will always be a direction that is 90 degrees to the plane of rotation, which is weird, but required for things like gyroscopic precession to make sense.