Rotation, Moment of Inertia, Torque and all that..

Here are three lectures related to a subject which for good reason has been troubling some students: From rotation of rigid bodies to angular Momentum, moment of Inertia and Torque.  What we usually call “Impulsmomentsætningen” (IMS) is in reality just Newton’s 2. law for rotation, so including Newton’s 2. law

we have the IMS, “Newton’s 2. law for rotation”:

Here is momentum (also called linear momentum), and is angular momentum, and is the moment of Inertia. Remember that the torque of a force is , i.e. a cross-product. In a mechanics problem with rotation, you usually would need both Newton’s 2′nd law and IMS to solve it.

Don’t forget “rolling without slipping” of a body with radius R. In this case, you have:

, and .

The kinetic energy of a body that rolls is,

,

the sum of the rotational kinetic energy and the translational kinetic energy.

— o —

Topics covered in this lecture:

Rotation of rigid bodies — Moment of Inertia — Parallel axis theorem

Topics covered in this lecture:

Angular momentum — Torque — Conservation of angular momentum

Topics covered in this lecture:

Torque — Oscillating bodies

Fictitious Force

A fictitious force (also called a pseudo force) is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame. So, a fictitious force is one that seems to exist because a frame of reference is accelerated. The forces may be treated algebraically like a real force, but the fictitious forces do not necessarily obey Newton’s first law.

One example is the centrifugal force, another one is the Coriolis force.

Wikipedia entry

Conservation of mechanical energy

Conservation of mechanical energy (potential energy + kinetic energy) discussed in a class given at MIT by Prof. Walter Lewin (course 8.01, Physics I: Classical Mechanics, fall 1999). Motion of car in a loop explained at 23:50, with a (deadly) amusing experiment at 45:40.

Topics covered in this lecture:

Work – Conservative forces – Potential energy – Kinetic energy – Mechanical energy

(Young and Freedman, chap. 6)

Atwood Machine

The Atwood machine was invented in 1784 by, yes Mr. Atwood. Two weights of mass and are connected by a string over a pulley. The acceleration is

Here is a demonstration from MIT, USA:

Homework: check that the experiment measures an acceleration as given by the equation above.

Circular motion

When an object moves in a circular orbit (of radius ), the acceleration has two components, tangential and radial. The angular speed is defined as , and angular acceleration defined as ( is the angle that measures the movement of the position vector); the two components are then generally:

In many cases we work with uniform circular motion, in this case .

credit: Prof. Walter Lewin of MIT (course 8.01, Physics I: Classical Mechanics, fall 1999).

Topics covered in this lecture:

Circular Motion – Centrifuges Moving – Reference Frames – Perceived Gravity

How to solve it

George Polya (1887-1985) was a Hungarian mathematician. He is famous for work on mathematics education, and for the 4 steps to solve a problem:

1. understand the problem

2. devise a plan

3. carry out the plan

4. look back

We can use this method to solve physics problems also. Try it! (It is a good idea to make a drawing of the situation in step 1). Single page explaining Polya’s principles:

how-to [PDF]

Video lectures: 3D Kinematics, projectile motion

credit: Prof. Walter Lewin of MIT (course 8.01, Physics I: Classical Mechanics, fall 1999).

Topics covered in this lecture:

Vectors — dot product — cross product — kinematics — projectile motion

(Young and Freedman, chap. 3)

Interactive simulation: Projectile motion

Projectile motion: footballs, humans and pianos – they all (ideally) move in the same way. Try the interactive simulation by following this link. The equation of motion is just

[the simulation is from University of Colorado at Boulder].

Equation summaries corrected

Equation summaries for 10020 and 10022 have been corrected to version 1.1:

eqsummary10020 [PDF]

eqsummary10022 [PDF]

HyperPhysics

“HyperPhysics” is not science fiction, its science: a nice and simple overview of mechanics, with links to and from different subjects. Try it.

HyperPhysics is hosted by the Department of Physics and Astronomy (Georgia State University).