Physics chapter 8

other names for toque

• angular force

• rotational force

Torque, T

the tendency of a force to rotate an object about some axis

torque formula

T = r F

• T = torque

• r = length of the position vector/lever arm/moment arm

• F = force

• r is the distance axis of rotation to F

SI unit for torque

Newton _^x meter (Nm)

Force vs torque

• forces cause accelerations

• torques cause angular accelerations

• force and torque are related

torque is a ___ quantity

vector

• direction is perpendicular to the plane determined by the position vector and the force

If the turning tendency of the force is counterclockwise, the torque will be

positive

If the turning tendency is clockwise, the torque will be

negative

Right Hand Rule

• Point the fingers in the direction of the position vector
• Curl the fingers toward the force vector
• The thumb points in the direction of the torque

What affects Torque?

• The wrench is free to rotate about an axis through O
• There are three factors that determine the effectiveness of the force in opening the door:
• The magnitude of the force
• The position of the application of the force
• The angle at which the force is applied

• changing r affects torque value

In torque, the applied force is not always

perpendicular to the position vector

• The component of the force perpendicular to the object will cause it to rotate

• W = fdcosΘ

• T = rFsinΘ

When the force is parallel to the position vector,

no rotation occurs

When the force is at some angle, the perpendicular component causes

the rotation

Taking the angle into account leads to a more general definition of torque:

T = r F sin Θ (given on exam)

• Θ = the angle between the force and the position vector

Net Torque

When two or more torques are acting on an object, the torques are added

• as vectors

If the net torque is zero, the object’s rate of rotation

doesn’t change

The net torque is the sum of

all the torques produced by all the forces

Counterclockwise torques are

positive

Clockwise torques are

negative

first condition of equilibrium

The net external force must be zero

• This is a statement of translational (linear) equilibrium

The Second Condition of Equilibrium states

The net external torque must be zero

• This is a statement of rotational equilibrium

torque math example

Moment of Inertia (aka Rotational Mass)

The angular acceleration is inversely proportional to the analogy of the mass in a
rotating system

• This mass analog is called the moment of inertia, I, of the object

Si units for moment of inertia

kg m²

Moment of Inertia is also known as

rotational inertia or angular mass

difference between moment of inertia and mass

the moment of inertia depends on the quantity of matter and its distribution in the rigid object

The moment of inertia also depends upon

the location of the axis of rotation

A majorette twirling a baton: Moment of Inertia,

The moment of Inertia of a system depends on,

how the mass is distributed and on the location of the axis of rotation

Moment of Inertia of a Uniform Ring

• Imagine the hoop is divided into a number of small segments, m1 ...
• These segments are equidistant from the axis

moment of inertia example for rotating object

When axis of rotation is outside the object rotating, the object is treated as

point mass

point mass formula

I_point-mass = mr²

Newton’s Second Law for a Rotating Object

• The angular acceleration is directly proportional to the net torque
• The angular acceleration is inversely proportional to the moment of inertia of the object

• like F = ma

Work done in rotational motion is

W = T 𝜃

• T = angular force

• 𝜃 = angular displacement

SI unit for work done in rotational motion

Joules (J)

Rotational Kinetic Energy

KE = ½ mv2
(since vt = rω)
KE = ½ mr2ω2 = ½ Iω2

• An object rotating about some axis with an angular speed ω, has rotational kinetic energy KEr = ½Iω2
• Energy concepts can be useful for simplifying the analysis of rotational motion

Work – Energy Theorem and Power in rotational motion

Angular Momentum

Conservation of Law of Angular Momentum (LCAM)

Applies to macroscopic objects as well as atoms and molecules

Law of Conservation of Angular Momentum, Example

With hands and feet drawn closer to the body, the skater’s angular speed increases
• L is conserved, I (=1/2mr²) decreases, w increases
𝐿 = 𝐼𝜔

Law of Conservation of Angular Momentum, Example, cont.

Coming out of the spin, arms and legs are extended and rotation is slowed
• L is conserved, I (=1/2mr2) increases, w decreases
𝐿 = 𝐼𝜔

in L = I w, if I increases

w decreases (and vice versa)

• in skating example, spreading hands out increases I which decreases w