Torque
ℓ
F
Axis
• τ = F·ℓ
• F and ℓ must be perpendicular.
• Units: N*m (enter them this way into computer)
A board is placed in a support. F1 and F2 push down on the board. If
the board stays stationary, how do the forces compare?
(1) F1 > F2
F1
F2
(2) F1 = F2
(3) F1 < F2
Support (fulcrum)
Clicker!
Which of the following forces provides the greatest torque about the given axis?
(the black dot represents the axis of rotation)
2F
2F
1.5ℓ
ℓ
A
F
B
(1) A
(2) B
|τA| = 2F*1.5ℓ = 3F*ℓ
|τB| = 2F*ℓ = 2F*ℓ
|τC| = F*2.5ℓ = 2.5F*ℓ
(3) C
2.5ℓ
(4) All the same.
C
Clicker!
Which of the following signs (+ or -) are true?
2F
2F
1.5ℓ
A
(1) - tA , - tB , + tC
ℓ
F
B
2.5ℓ
τA : CW
(-)
(2) + tA , - tB , - tC
(3) - tA , + tB , + tC
τA : CCW (+)
τA : CCW (+)
Sign Convention: (+) for CCW rotation; (-) for CW
C
Levers
Small force with long lever arm
can exert large force with short
level arm
Class 1 Lever – Fulcrum
between input effort and
output load
• Pliers
• Catapult
• Oars
• Crowbar
• Seesaw
Equilibrium
• Equilibrium – Accel = 0
• Static Equilibrium – Accel = 0 AND velocity=0
– Stationary
What happens?
What happens?
Example.
A uniform board of 10 m is supported at the center. If a person (80
kg mass) is standing 2 m away from the center, how much force is
required to balance the board at the other end?
F1
F2
Support (fulcrum)
Clicker!
A force of 5N is applied downward 0.3m to the left of the fulcrum.
Where would one exert a force of 1 N downward to balance the the
stick?
5N
(1) 0.2m to the left
(2) 0.2m to the right
(3) 1.5 m to the left
(4) 1.5 m to the right
0.3 m
Support (fulcrum)
Στ = 0
(5N)(0.3m) - (1N)L = 0
L = 1.5m
Clicker!
A downward force of 5N is applied to a uniform bar 0.3m to the left of the fulcrum.
Another force of 1 N is applied downward 1.5 m to the right of the fulcrum. How much
force is applied by the fulcrum to the bar? The bar is in equilibrium.
5N
(1) 2 N.
(2) 3 N.
(3) 6 N
(4) 3.5N
0.3 m
1.5 m
Support (fulcrum)
ΣFy = 0
F - 5N - 1N = 0
F = 6N
1N
Clicker!
At which point and at which direction would the least amount of force be required
to hold the lever stationary?
Longest lever arm - perpendicular force
Clicker!
Two symmetric objects: a uniform bar and a square box.
Where do you draw the weight?
A
1)
2)
3)
4)
5)
A, C,
B, D
C, E, F
B, E
A, C, D, F
B
C
D
E
F
Clicker!
An asymmetric object. Where do you think the weight will be
centered?
1)
2)
3)
4)
A
B
C
D
A
B
C
D
Center of Gravity (Center of Mass)
•
•
In extended objects, the location where the weight is considered acting on it is
the center of gravity.
If the object is symmetric, then the "real center“ of the object is the center of
gravity.
center of gravity
•
If irregular, one can find the center of gravity using the formula:
x cg 
W 1 x1  W 2 x 2  
W1  W 2  
Center of Gravity
An asymmetric object.
1
x1
x cg 
2
3
x2
W 1 x1  W 2 x 2  
W1  W 2  
4
Clicker!
Roughly where is the center of gravity located? Point 1, 3, or 5 or region 2
(between 1 and 3) or 4 (between 3 and 5)?
x cg 
x cg 
W 1 x1  W 2 x 2  
W1  W 2  
4 kg  g  0 m  2 kg  g  1m  2 kg  g  2 m
 6 / 8  0 . 75
4 kg  g  2 kg  g  2 kg  g
Clicker!
F
What are the forces do you need to consider if the bar
is stationary?
q
Hinge
q
Fsinq
F
W (1)
Fsinq
W (2)
Fcosq
SFx = 0, and SFy = 0
W (3)
Torque
If the F is not perpendicular to ℓ , then find the F component
perpendicular to ℓ .
F
Fsin q
q
ℓ
Hinge
t = (F sin q) . ℓ
Note: Check where the angle is given. It could be Fcos q, if
the angle is given in other side.
Clicker!
What is the torque on the fishing pole? Use the end where it is held as the axis of
rotation.
(1) 80N (cos 20°) 1.2m
(2) 80N (sin 20°) 1.2m
(3) 80N*1.2m
(4) -80N (cos 20°) 1.2m
(5) -80N (sin 20°) 1.2m
(6) -80N*1.2m
Clicker!
A cook holds a 2.00kg carton of milk at arm's length. The bicep muscle exerts a
force Fb on the forearm. What is the expression for the torque of the bicep muscle
on the forearm? Use the elbow as the axis of rotation.
(1) (Fbcosθ) 8cm
(2) (Fbsinθ) 8cm
(3) Fb 8cm
(4) -(Fbcosθ) 8cm
(5) -(Fbsinθ) 8cm
(6) -Fb 8cm
(7) None of the above
Clicker!
The bridge shown must be in static equilibrium. A truck and a car are crossing the
bridge. Which of the following force diagrams most correctly describe the forces?
(1)
(2)
(3)
(4)
Example
Write the equation for the net torque.
F
FT
W
Fc
Clicker!
(1)
What is the correct force diagram? Assume the
gray line as the person.
(2)
(3)
(4)