Torque

We know that Newton’s second law ( ) explains that the net force is the source of an object’s acceleration.

What is the source of a rotating object’s angular acceleration? It can’t be just a force, because it matters where on the object that force is applied.

The answer lies in the quantity called torque.

F ma

Torque…

Torque, , is the tendency of a force to rotate an object about some axis Torque is a vector = r F sin = F d

F is the force is the angle the force makes with the horizontal d is the moment arm (or lever arm)

…Torque…

The moment arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin Φ

…Torque

The horizontal component of F (F cos ) has no tendency to produce a rotation

Torque will have direction 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

You are trying to open a door that is stuck by pulling on the doorknob in a direction perpendicular to the door. If you instead tie a rope to the doorknob and then pull with the same force, is the torque you exert increased?

A.yes

B.no

Conceptest…

You are trying to open a door that is stuck by pulling on the doorknob in a direction perpendicular to the door. If you instead tie a rope to the doorknob and then pull with the same force, is the torque you exert increased?

A.yes

B.no

…Conceptest

You are using a wrench and trying to loosen a rusty nut. Which of the arrangements shown is most effective in loosening the nut? List in order of descending efficiency the following arrangements:

                                                                 

Conceptest…

You are using a wrench and trying to loosen a rusty nut. Which of the arrangements shown is most effective in loosening the nut? List in order of descending efficiency the following arrangements:

                                                                 

…Conceptest

A plumber pushes straight down on the end of a long wrench as shown. What is the magnitude of the torque he applies about the pipe at lower right?

A. (0.80 m)(900 N)sin 19°

B. (0.80 m)(900 N)cos 19°

C. (0.80 m)(900 N)tan 19°

D. none of the above

Conceptest…

A. (0.80 m)(900 N)sin 19°

B. (0.80 m)(900 N)cos 19°

C. (0.80 m)(900 N)tan 19°

D. none of the above

A plumber pushes straight down on the end of a long wrench as shown. What is the magnitude of the torque he applies about the pipe at lower right?

…Conceptest

Net Torque

The force F1 will tend to cause a counterclockwise rotation about O

The force F2 will tend to cause a clockwise rotation about O

F1d1 – F2d2

Torque vs. Force Forces can cause a change in linear

motion Described by Newton’s Second Law

Forces can cause a change in rotational motion The effectiveness of this change depends on

the force and the moment arm The change in rotational motion depends on

the torque

Torque Units

The SI units of torque are N.m Although torque is a force multiplied by a

distance, it is very different from work and energy The units for torque are reported in N.m and not

changed to Joules

Torque and Angular Acceleration, Wheel Example The wheel is rotating

and so we apply The tension supplies the

tangential force

The mass is moving in a straight line, so apply Newton’s Second Law Fy = may = mg - T

Torque and Angular Acceleration, Multi-body Ex.,

1 Both masses move in linear directions, so apply Newton’s Second Law

Both pulleys rotate, so apply the torque equation

Torque and Angular Acceleration, Multi-body Ex.,

2

The mg and n forces on each pulley act at the axis of rotation and so supply no torque

Apply the appropriate signs for clockwise and counterclockwise rotations in the torque equations

Problem

A model airplane with mass 0.750 kg is tethered by a wire so that it flies in a circle 30.0 m in radius. The airplane engine provides a net thrust of 0.800 N perpendicular to the tethering wire.

(a) Find the torque the net thrust produces about the center of the circle.

(b) Find the angular acceleration of the airplane when it is in level flight.

(c) Find the linear acceleration of the airplane tangent to its flight path.

Answers

a) 24.0 N-m

b) 0.0356 rad/s2

c) 1.07 m/s2

Review: The Vector Product

Given two vectors, A and B

The vector (“cross”) product of A and B is defined as a third vector, C

C is read as “A cross B”

The magnitude of C is AB sin is the angle between A and B

More About the Vector Product

The quantity AB sin is equal to the area of the parallelogram formed by A and B

The direction of C is perpendicular to the plane formed by A and B

The best way to determine this direction is to use the right-hand rule

Properties of the Vector Product

The vector product is not commutative. The order in which the vectors are multiplied is important

To account for order, rememberA x B = - B x A

If A is parallel to B ( = 0o or 180o), then A x B = 0 Therefore A x A = 0

Vector Products of Unit Vectorsˆ ˆ ˆ ˆ ˆ ˆ 0

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆˆ ˆ

i i j j k k

i j j i k

j k k j i

k i i k j

Signs are interchangeable in cross productsA x (-B) = - A x B

ˆ ˆ ˆ ˆ i j i j

The Vector Product and Torque

The torque vector lies in a direction perpendicular to the plane formed by the position vector and the force vector

= r x F

The torque is the vector (or cross) product of the position vector and the force vector

Torque Vector Example

Given the force

= ?

m)ˆ00.5ˆ00.4(

N)ˆ00.3ˆ00.2(

jir

jiF

ˆ ˆ ˆ ˆ [(4.00 5.00 )N] [(2.00 3.00 )m]

ˆ ˆ ˆ ˆ[(4.00)(2.00) (4.00)(3.00)

ˆ ˆ ˆ ˆ(5.00)(2.00) (5.00)(3.00)

ˆ2.0 N m

r F i j i j

i i i j

j i i j

k