Welcome back to Physics 211

1
Welcome back to Physics 211

• Todays agenda
• Torque
• Rotational Motion

2
Extended objectsneed extended free-body diagrams

• A point free-body diagram allows finding net
  force since points of application do not matter.
• Extended free-body diagrams show point of
  application for each force and allow finding net
  torque.

3
Conditions for equilibrium of an extended object

• For an extended object that remains at rest and
  does not rotate

• The net force on the object has to be zero.
• The net torque on the object has to be zero.

4
Restatement of equilibrium conditions

• m1r1m2r20 ? W1r1W2r20
• i.e S force (W) x displacement (r) 0
• The quantity force x displacement is
• called torque (more shortly)
• Thus, equilibrium requires the net torque to be
  zero

5
EXAMPLE A 1-kg mass is fastened to a meter stick
near one end. A person balances the system by
placing a finger directly below point P which is
just to the left of the mass. Is the center of
mass of the system located

• 1. to the left of point P,
• 2. at point P, or
• 3. to the right of point P?
• 4. Unable to decide.

6
Tentative definition of torque
The torque on an object with respect to a given
pivot point and due to a given force is defined
as the product of the force exerted on the object
and the moment arm. The moment arm is the
perpendicular distance from the pivot point to
the line of action of the force.
7
EXAMPLE A meterstick is pivoted at its center of
mass. It is initially balanced. A mass of 200
g is then hung 20 cm to the right of the pivot
point. Is it possible to balance the meter-stick
again by hanging a 100-g mass from it?

• 1. Yes, the 100-g mass should be 20 cm to the
  left of the pivot point.
• 2. Yes, but the lighter mass has to be farther
  from the pivot point (and to the left of it).
• 3. Yes, but the lighter mass has to be closer to
  the pivot point (and to the left of it).
• 4. No, because the mass has to be the same on
  both sides.

8
The Leaning Tower demo

• Tower does not fall if the vertical line from its
  CM lies within base area

9
Reason

• There are two external forces on the tower
• 1) Its weight W This force effectively goes
  through the CM of the tower. If we choose the
  pivot point P at the base edge the weight force
  results in a torque about P.
• 2) Normal Force N The normal force N of the
  ground on the tower. The force N is upward and on
  the opposite side of P relative to the weight
  force W.
• The torque created by N reinforces the torque
  created by W. If there is a net torque,
  equilibrium is not possible

10
Computing torque
F

• tFd
• Frsinq
• (F sinq)r
• Note that F sin?
• is the component
• of the force at 900
• to position vector
• times distance

q
r
d
O
11
Interpretation of torque

• Measures tendency of any force to cause rotation
• Torque is defined with respect to some origin
  must talk about torque of force about point X
  etc
• Torques can cause clockwise () or anticlockwise
  rotation (-)

12
demo fighting torque!

• hold bar add weights at different distances
• effort increases with distance and magnitude of
  weight force

13
Conditions for equilibrium of an extended object

• For an extended object that remains at rest and
  does not rotate

• The net force on the object has to be zero.
• The net torque on the object has to be zero.

14
What if t not zero ?

• If the torque about some pivot point is not zero,
  the object will rotate about the pivot.
• Rotation is consistent with direction of force

15
A T-shaped board is supported such that its
center of mass is to the right of and below the
pivot point. Which way will it rotate?
CM

• 1. Clockwise.
• 2. Counter-clockwise.
• 3. Not at all.
• 4. Not sure what will happen.

16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
Rotations about fixed axis

• Every particle in a rigid body undergoes circular
  motion (not necessarily constant speed) with the
  same time period
• v(2pr)/Tw r. Quantity w is called angular
  velocity
• Similarly can define angular acceleration aDw/Dt

20
Vector (or cross) product of vectors
The vector product is a way to combine two
vectors to obtain a third vector that has some
similarities with multiplying numbers. It is
indicated by a cross (?) between the two
vectors. The magnitude of the vector cross
product is given by The direction of the
vector A?B is perpendicular to the plane of
vectors A and B and given by the right-hand rule.
21
Right Hand Rule

• To get the direction of A x B do the following
• Put the fingers of your right hand in the
  direction of A
• Then curl these fingers toward the direction of
  B
• Then outstretch your right thumb and its
• direction is that of A x B

22
REMINDER Scalar (or dot) product of vectors
The scalar product is a way to combine two
vectors to obtain a number (or scalar) that has
some similarities with multiplying numbers (i.e.,
a product). It is indicated by a dot () between
the two vectors.
23
Definition of torque
where r is the vector from the reference point
(generally either the pivot point or the center
of mass) to the point of application of the force
F.
where q is the angle between the vectors r and F.
24

• The definition of torque
• T r x F
• is our first application of the general concept
  of the cross product.
• Previously, we have utilized an application of
  the dot product in defining the concept of work W
  produced by a force F
• W F. r
• where r is the displacement
• There are many applications of both types of
  products