Unit 3, Chapter 9

CPO ScienceFoundations of Physics

Chapter 9Chapter 9

Unit 3: Motion and Forces in 2 and 3 Dimensions

9.1 Torque

9.2 Center of Mass

9.3 Rotational Inertia

Chapter 9 Torque and Rotation

Chapter 9 Objectives

Calculate the torque created by a force. Solve problems by balancing two torques in

rotational equilibrium. Define the center of mass of an object. Describe a technique for finding the center

of mass of an irregularly shaped object. Calculate the moment of inertia for a mass

rotating on the end of a rod. Describe the relationship between torque,

angular acceleration, and rotational inertia.

Chapter 9 Vocabulary Terms torque

center of mass

angular acceleration

rotational inertia

rotation

translation

center of rotation

rotational equilibrium

lever arm

center of gravity

moment of inertia

line of action

9.1 Torque

Key Question:

How does force create rotation?

*Students read Section 9.1 AFTER Investigation 9.1

9.1 Torque

A torque is an action that causes objects to rotate.

Torque is not the same thing as force.

For rotational motion, the torque is what is most directly related to the motion, not the force.

9.1 Torque

Motion in which an entire object moves is called translation.

Motion in which an object spins is called rotation.

The point or line about which an object turns is its center of rotation.

An object can rotate and translate.

9.1 Torque

Torque is created when the line of action of a force does not pass through the center of rotation.

The line of action is an imaginary line that follows the direction of a force and passes though its point of application.

9.1 Torque

To get the maximum torque, the force should be applied in a direction that creates the greatest lever arm.

The lever arm is the perpendicular distance between the line of action of the force and the center of rotation

9.1 Torque

= r x F

Lever arm length (m)

Force (N)

Torque (N.m)

9.1 Calculate a torque

A force of 50 newtons is applied to a wrench that is 30 centimeters long.

Calculate the torque if the force is applied perpendicular to the wrench so the lever arm is 30 cm.

9.1 Rotational Equilibrium When an object is in rotational equilibrium, the net torque applied to

it is zero.

Rotational equilibrium is often used to determine unknown forces.

What are the forces (FA, FB) holding the bridge up at either end?

9.1 Rotational Equilibrium

9.1 Calculate using equilibrium

A boy and his cat sit on a seesaw.

The cat has a mass of 4 kg and sits 2 m from the center of rotation.

If the boy has a mass of 50 kg, where should he sit so that the see-saw will balance?

9.1 When the force and lever arm are NOT perpendicular

9.1 Calculate a torque

It takes 50 newtons to loosen the bolt when the force is applied perpendicular to the wrench.

How much force would it take if the force was applied at a 30-degree angle from perpendicular?

A 20-centimeter wrench is used to loosen a bolt.

The force is applied 0.20 m from the bolt.

9.2 Center of Mass

Key Question:

How do objects balance?

*Students read Section 9.2 AFTER Investigation 9.2

9.2 Center of Mass There are three different axes about which an object will naturally spin. The point at which the three axes intersect is called the center of mass.

9.2 Finding the center of mass If an object is irregularly shaped, the center of mass can be found by

spinning the object and finding the intersection of the three spin axes. There is not always material at an object’s center of mass.

9.2 Finding the center of gravity

The center of gravity of an irregularly shaped object can be found by suspending it from two or more points.

For very tall objects, such as skyscrapers, the acceleration due to gravity may be slightly different at points throughout the object.

9.2 Balance and center of mass

For an object to remain upright, its center of gravity must be above its area of support.

The area of support includes the entire region surrounded by the actual supports.

An object will topple over if its center of mass is not above its area of support.

9.3 Rotational Inertia

Key Question:

Does mass resist rotation the way it resists acceleration?

*Students read Section 9.3 AFTER Investigation

9.3

9.3 Rotational Inertia Inertia is the name for an

object’s resistance to a change in its motion (or lack of motion).

Rotational inertia is the term used to describe an object’s resistance to a change in its rotational motion.

An object’s rotational inertia depends not only on the total mass, but also on the way mass is distributed.

9.3 Linear and Angular Acceleration

a = rRadius of motion

(m)

Linear acceleration

(m/sec2)

Angular acceleration (kg)

9.3 Rotational Inertia To put the equation into rotational motion variables, the force is

replaced by the torque about the center of rotation.

The linear acceleration is replaced by the angular acceleration.

9.3 Rotational Inertia A rotating mass on a

rod can be described with variables from linear or rotational motion.

9.3 Rotational Inertia

The product of mass × radius squared (mr2) is the rotational inertia for a point mass where r is measured from the axis of rotation.

9.3 Moment of Inertia

The sum of mr2 for all the particles of mass in a solid is called the moment of inertia (I).

A solid object contains mass distributed at different distances from the center of rotation.

Because rotational inertia depends on the square of the radius, the distribution of mass makes a big difference for solid objects.

9.3 Moment of Inertia

The moment of inertia of some simple shapes rotated around axes that pass through their centers.

9.3 Rotation and Newton's 2nd Law If you apply a torque to a wheel, it will spin in the direction of the torque.

The greater the torque, the greater the angular acceleration.

Application: Bicycle Physics