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Torque
and
Equilibrium
Lecture 8 Pre-reading: KJF §8.1 and 8. 1
Archimedes’ Lever Rule
At equilibrium (and with forces 90° to lever): r 1 F 1 = r 2 F 2 2 2 3
General Lever Rule
For general angles r 1 F 1 sin! 1 = r 2 F 2 sin! 2 We call rF sin! = "# torque S.I. unit of torque : newton metre (Nm) At equilibrium, the magnitude of torques exerted at each end of lever are equal KJF §7. 3 Crudely speaking, torque is "twisting or turning ability" of a force that can:
- (^) change the angular velocity of an object (i.e. speed up or slow down rotation)
- (^) cause a twisting or bending distortion of an object A force with a "line of action" that does not cross the axis of rotation results in torque.
What is torque?
4 4 Note:
- (^) torque is measured about a particular point. Usually this will be a hinge, pivot or axis
- (^) torque has a sign. All forces that tend to rotate the object in the same direction produce torque with the same sign 5 6
Calculating torque (1)
Choose a sign convention (e.g. anti-clockwise +ve), then decide in which direction force is pulling or pushing lever. Write that sign in front of your answer. Method 1: If you're given r and !, use formula for torque (magnitude)
" = r$F sin!
(Note: sin! = sin%,! it doesn’t matter which angle you use)
Example : Calculate torque on lever exerted by hand:
7 Calculating torque (2) Method 2:
If you're given d the “perpendicular distance”
from axis to the “line of action”, then use formula
" = d F
If the “line of action” crosses the axis
(i.e. d $=$0) then " = 0 7 8 Opening a door
- (^) If r is perpendicular to F, then torque " = r F
- (^) If r is not perpendicular to F, then torque " = r F sin! where! is the angle between r and F Axis of rotation r F
- (^) What happens if you push in the middle of the door; do you need more or less force? Why?
- (^) What happens if you push along a line passing through axis of rotation? Explain. 8 9 Problem The length of a bicycle pedal arm is r $=$0.152$m, and a downward force of F $=$ 111 $N is applied by the foot. What is the magnitude of torque about the pivot point when the angle! between the arm & vertical is; (a) 30.0°? (b)# 90.0°? (c) 180.0°? [8.44 Nm, 16.9 Nm, 0.00 Nm] 9 10 Adding up Torques We will only consider torques acting in 2D (flat on page)
- (^) Choose a sign convention (e.g. anti-clockwise is positive).
- Choose the rotation axis around which to calculate torque (unless it's already given).
- (^) Draw the line of action for each force
- (^) For each force, calculate the resulting torque (including sign).
- Add up all the torques. KJF §7.2, see p. 214 10 11
F 1
F 2
12 Adding up Torques: Example
torque 1; " 1 = – rF 1 sin! = –0.5 & 10 & sin 30 = –2.50 Nm
torque 2; " 2 = + mgd = 1 & 9.8 & 0.25 = +2.45 Nm
#! net torque = '" = " 1 + " 2 = 2.45 + (–2.50) = –0.05 Nm (i.e. clockwise)
19
But...
Too much cargo is loaded at the back. If the wheel is chosen as the rotation axis, all resulting torques are acting in the clockwise direction. There is no torque opposing the torque due to the weight of the system, hence there is a net clockwise torque. The system will rotate until the cart hits the ground. The donkey will be lifted off the ground. 19 20 20 21 21 22
Types of Equilibrium
Neutral: with a small displacement, remains at new position. Stable: with a small displacement, returns to original position. Unstable: with a small displacement, continues to move away from equilibrium position. KJF §8. 22 23
Stable
W
N
net torque around X X 24 W
N
NO net torque around X X
Neutral
25 W
N
net torque around X X
Unstable
25 26 26
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Momentum, impulse and energy Read: KJF §9.1, 9.
