Measuring Rotational Motion

Chapter 7 Section 1

What is Rotational Motion?

Rotational Motion – The motion of a body that spins about an axis.

Examples:Ferris WheelBicycle WheelMerry-go-roundEtc…

Circular Motion

A point that rotates about a single axis undergoes circular motion around that axis.

Regardless of the shape of the object, any single point on the object travels in a circle around the axis of rotation.

Linear motion here? Think again

Its difficult to describe the motion of a point in a circular path through linear quantities, because the direction of the motion is not in a straight line path.

Therefore it is describe in terms of angle through which the point on an object moves.

Rotational Motion

When rotational motion is described using angles, all points on a ridged rotating object, except the points on the axis move through the same angle during any time interval.

r = radius θ = angle s = arc length

Reference Liner

θs

Radians

Radian – An angle whose arc length is equal to its radius, which is approximately equal to 57.3º

The radian is a pure number with no dimensions.

Radian Chart

Radian Equations

Any angle θ, measured in radians, is defined as the following:

To convert degrees to radians, or vise versa, use the following equation:

Angular Displacement

Angular Displacement – The angle through which a point, line, or body is rotated in a specific direction and about a specified axis.

In simple terms, angular displacement describes how much an object has rotated. It is not a distance!

Angular Displacement Equation

Δθ = Angular Displacement (rad) Δs = Change in Arc Length (m) r = Distance from Axis (m)

When Δθ rotates: Clockwise – Negative Counter-clockwise – Positive

Recall

Recall that Δ is a capital Greek letter, “delta”

It means, “Change in” It is always the final minus the initial

Example: Δθ = θf-θi

Example Problem Earth has an equatorial radius of

approximately 6380km and rotates 360º in 24 hours.

1. What is the angular displacement (in degrees) of a person standing on the equator for 1.0 hour?

2. Convert this angular displacement to radians

3. What is the arc length traveled by this person?

Example Problem Answer

1. 15º

2. 0.26 rad

3. Approximately 1700km

Example Problem

Earth has an average distance from the sun of approximately 1.5x10^8 km. For its orbital motion around the sun, find the following:

1. Average daily angular displacement

2. Average daily distance traveled

Example Problem Answer

1. 1.72x10^-2 rad

2. 2.58x10^6 km

Angular Speed Angular Speed – The rate at which a body

rotates about an axis, usually expressed in radians per second. (rad/s)

Recall that linear speed describes the distance traveled in a specific time, here it is the amount of rotation in a specific time. It describes how quickly the rotation occurs.

Angular Velocity Equation

ω = Angular Velocity (rad/s)Lower case Greek letter called, “omega”

Δθ = Angular Displacement (rad) t = Time Interval (s)

Revolutions

Occasionally angular speeds are given in revolutions per unit time.1 rev = 2πrad

Examples:DVD’sRecordsEngines

2000 rpm (revolutions per minute)

Example Problem

An Indy car can complete 120 laps in 1.5 hours on a circular track. Calculate the average angular speed of the Indy car.

Example Problem Answer

0.14 rad/s

Angular Acceleration

Angular Acceleration – The time of change of angular speed, expressed in radians per second per second.

Angular Acceleration Equation

α = Angular Acceleration (rad/s2)Lower case Greek letter, “alpha”

Δω = Change in Angular Velocity t = Time Interval

Example Problem

A yo-yo at rest is sent spinning at an angular speed of 12 rev/s in 0.25 seconds. What is the angular acceleration of the yo-yo?

Example Problem Answer

301.6 rad/s²

All Points Have Same Speed and Acceleration If a point on the rim of a bicycle wheel had

a greater angular speed than a point closer to the axel, the shape of the wheel would be changing.Which can not happen in normal everyday

riding. All points on a rotating object have the

same angular speed & acceleration.

Comparing Linear and Angular Quantities

Angular Substitutes for Linear Quantities

Angular Linear

Displacement θ (units: rad) d (units: m)

Speed ω (units: rad/s) v (units: m/s)

Acceleration α (units: rad/s²) a (units: m/s²)

Kinematic Equations Can be Used for Circular Motion The same equations used for linear motion

with constant acceleration, can be used for circular motion with constant angular acceleration.

Change the variables, by using the table in the previous slide and you have the new kinematic equations.

Kinematic Equations for Constant Angular Acceleration

Angular Linear

222

221

if

i

if

tt

t

advv

attvd

atvv

if

i

if

222

221

Example Problem

A barrel is given a downhill rolling start of 1.5rad/s at the top of a hill. Assume a constant angular acceleration of 2.9rad/s²

1. If it takes 11.5s to get to the bottom of the hill, what is the final angular speed of the barrel?

2. What angular displacement does the barrel experience during the 11.5s ride?

Example Problem Answers

1. 35 rad/s

2. 210 rad