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Notes: Torque and Static Equilibrium
Sections 7.8, 8.1, 8.4, 8.5, 9.1, 9.2, & 9.4
A circle is just a straight line rolled up.
Rotational Quantities
If you spin a wheel, and look at how fast a point on the wheel is spinning, the answer depends on how far away
the point is from the axle. Velocity, then, isn't the most convenient thing to use when you're dealing with
rotation, and for the same reason neither is displacement, or acceleration; it is often more convenient to use their
rotational equivalents.
The equivalent variables for rotation are angular displacement (θ)(angle, for short); angular velocity (ω), and
angular acceleration (α). All the angular variables are related to the straight-line variables by a factor of r, the
distance from the center of rotation to the point you're interested in.
Although points at different distances from the center of a rotating wheel have different velocities, they all have
the same angular velocity. Angles (angular displacements, that is) are generally measured in radians, and
therefore all other angular quantities are expressed in radians as well.
Any equation dealing with rotation can be found from its straight-line motion equivalent by substituting the
corresponding rotational variables. The rotational kinematics equations apply when the angular acceleration is
constant. The equations are the same as the constant-acceleration equations for 1-D motion, substituting the
rotational equivalents of the straight-line motion variables.
v = vo + at ………………….. ________________
d = vot + ½ at2 ……………....________________
d = ½ (v + vo) ……………… ________________
v2 = vo2 + 2ad ……………… ________________
Center of Gravity
For a given body, the center of mass is the average location of all the mass that makes up the object. A
symmetrical object like a ball can be thought of as having all of its mass concentrated at its geometric center; by
contrast, an irregularly shaped object such as a baseball bat has more of its mass toward one end. A solid cone
has its center of mass exactly one fourth of the way up from its base.
The center of gravity of an object is the point you can suspend the object from without there being any rotation
because of the force of gravity, no matter how the object is oriented. If you suspend an object from any point,
let it go and allow it to come to rest, the center of gravity will lie along a vertical line that passes through the
point of suspension. Center of gravity is the same thing as the center of mass, except specifically referring to an
object under the influence of gravity. The terms are effectively synonymous, and we will use the abbreviation
CG for short, when consideration of this position is necessary.
The CG of a uniform but elogated object, such as a meter stick, is at its geometric center, for the stick acts as
though its entire weight were concentrated there. Support at that single point supports the whole stick.
Balancing an object provides a simple method of locating its CG.
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The CG of any freely suspended object lies directly beneath (or at) the point of support. If a vertical line is
drawn through the point of suspension, the CG lies somewhere along that line. To determine exactly where it
lies along the line, we have only to suspend the object from some other point and draw a second vertical line
through that point of suspension. The CG lies where the two lines intersect.
The CG may be a point where no matter exists. For example, the center of mass of a ring or a hollow sphere is
at the geometrical center.
For any object, the x-position of the center of gravity can be calculated by considering the weights and xpositions of all the pieces making up the object:
A similar equation would allow you to find the y position of the center of gravity.
Fact 1 - An object thrown through the air may spin and rotate, but its center of gravity will follow a smooth
parabolic path, just like a ball.
Fact 2 - If you tilt an object, it will fall over only when the CG lies outside the supporting base of the object.
Fact 3 - If you suspend an object so that its center of gravity lies below the point of suspension, it will be stable.
It may oscillate, but it won't fall over.
Torque
Translational motion is the movement of a particle along a path, where the position of the force on the particle
does not affect the motion of the particle. Rotational motion considers rigid bodies rather than particles, where
the position of the applied force has a large effect on the rotation of the object.
In the diagram to the right, F1 might cause some translational
motion of the beam but probably little rotation. F2 might cause
some translational motion of the beam but more likely, the beam
will rotate. What would happen if F2 were moved closer to the end
of the beam? Would it beam easier or harder to rotate?
F2
F1
Torque is the measure of the tendency of a force to rotate a body around some point, called the fulcrum or pivot.
The force is applied in such a way that it could cause rotation of an object if it is not balanced.


τ= Fr sinθ
Mathematically, it is defined as the product of the applied force and the
perpendicular distance from some point (typically measured from the
fulcrum). This length measurement is often called the "lever arm” or
“moment arm".
Only the perpendicular component of the force effects the rotation of the
lever. Torque is maximum when r is large, and sinθ = 1, so θ is 90º.

The units for torque are usually Nm (NOT the same as a Joule!)

Direction of torque is usually described as clockwise or counterclockwise with respect to the fulcrum.
The Second Condition of Equilibrium
A net torque, Στ, is analogous to a net force (ΣF) when examining its influence on motion. Στ applies to
rotational motion the same way ΣF applies to linear motion.
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If something is at equilibrium, that means that in addition to ΣF = 0, there is the condition
that Στ = 0.
All of the torques that cause the object to rotate clockwise must be balanced by all of the torques that cause the
object to rotate counterclockwise. Στcw + Στccw = 0. This means that the clockwise torques are equal to the
counterclockwise torques, Στcw = Στccw.
A few important notes about torque:
 Any torque that would try to cause counterclockwise rotation is considered to be positive.
 The exact location of where a force acts now becomes very important.
 The weight of an object seems to act through its center of mass.
 The words "balanced" or "equilibrium" or "at rest" imply Στ = 0.
An object at equilibrium has no net force acting on it, and has no net torque acting on it.
To solve Torque problems:
Draw a sketch.
Label the object with given information.
Label the weight of the object as a force at the center of gravity (generally the exact middle of the object).
Choose a fulcrum. NOTE: If the object is in equilibrium, it does not matter where you put the axis of rotation
for calculating the net torque; the location of the axis is completely arbitrary.
Show the direction of clockwise motion with an arrow.
Write the equation, Στcw = Στccw.
Write the sum of the clockwise torques, as F1·r1+ F2·r2 + F3·r3, etc….
Repeat for counterclockwise torques.
Substitute in any numbers that you have.
Solve for the unknown.
Rotational Inertia (moment of inertia)
Just as an object at rest tends to stay at rest, and an object in motion tends to remain moving in a straight line, an
object rotating about an axis tends to remain ROTATING about the same axis unless interfered with by some
external influence. The property of an object to resist changes in rotation is called rotational INERTIA. Things
that rotate tend to remain rotating, while non-rotating things tend to remain non-rotating.
Symbol: I, kgm2
Like linear inertia, rotational inertia of an object depends on its MASS. The greater the mass the
GREATER the rotational inertia.
Unlike linear inertia, rotational inertia of an object depends on the POSITION of the mass. The greater
the distance between the bulk of an object’s mass and its axis of rotation, the GREATER the rotational
inertia.
In general, the moment of inertia of a rigid body will be given in any problem you might be asked to answer.
Note: The units of moment of inertia are kgm2.
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Note: The complete set of dynamical equations needed to describe the motion of a rigid body consists of the
torque equation given above, plus Newton's Second Law applied to the center of mass of the object:
The moment of inertia, like torque, must be defined about a particular axis. It is different for different choices of
axes. Extended objects can again be considered as a very large collection of much smaller masses glued
together to which the definition of moment of inertia given above can be applied.
Examples of Moments of Inertia of Extended Objects:
uniform hoop: I = mr 2
cylindrical shell I = (1/12) mr 2
long thin rod (about middle) I = mL 2
long thin rod (about one end) I =⅓ mL 2
solid cylinder I = ½ mr 2
solid sphere I = (2/5)mr 2
The moment of inertia depends on how the mass is DISTRIBUTED about the axis. For a given total mass, the
moment of inertia is greater if more mass is FARTHER from the axis. An object where the mass is concentrated
close to the axis of rotation is EASIER to spin.
Rotational Kinetic Energy
A spinning object has rotational kinetic energy. The kinetic energy of rotation of a rigid body is again analogous
to the translational KE. A rolling object has both translational and rotational kinetic energy. Remember: The
law of conservation of energy holds in all situations.
Angular Momentum
Rotating things, whether a colony in space, a cylinder rolling down an incline, or an acrobat doing a somersault,
keep on rotating until something stops rotation. A rotating object has an “inertia of rotation”. This is called
Angular Momentum. It depends upon Rotational Inertia and rotational velocity. The faster something is
rotating, the more angular momentum it has. The harder it was to get it started rotating (the greater its inertia),
the more angular momentum it has.
A planet orbiting the sun, a rock whirling at the end of a string, and the tiny electrons whirling about atomic
nuclei all have angular momentum.
Just as an external net force is required to change the inertia of an object, an external net torque is required to
change the angular momentum of an object.
Angular momentum is conserved if no net torque acts on the system. This means that if the rotational inertia
increases, then the rotational velocity must decrease, (or vice versa) in order to maintain a constant angular
momentum. Because angular momentum is proportional to the moment of inertia, which depends on not just the
mass of a spinning object, but also on how that mass is distributed relative to the axis of rotation, some
interesting effects can be observed.
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Questions and Problems:
1. Where is the CG of the earth’s atmosphere?
2. Why is it dangerous to slide open the top drawers of a fully loaded file cabinet or dresser that is not
secured to the floor or wall?
3. When a car drives off a cliff, why does it rotate forward as it falls?
4. Why doesn’t the Tower of Pisa fall over?
5. How can you design objects to reduce the likelihood of tipping?
6. If you wish to have maximum speed a the very bottom of a roller coaster ride, should you sit in a front
car, a middle car, or a rear car?
7. A weight of 2 N is placed 0.2 m from the pivot of a 0.5-N beam. If the beam is 1-m long and the pivot is
in the exact center, where should you place a 1.5 N weight to balance the beam?
8. A weight of 2 N is placed 0.2 m from the pivot of a 0.5-N beam. If the beam is 1-m long and the pivot is
in the exact center, how much weight should be placed at 0.4 m from the pivot to balance the beam?
9. A weight of 2 N is placed 0.2 m from the pivot of a 0.5-N beam. If the beam is 1-m long and the pivot is
at the 0.3-m mark, where should you place a 1.5 N weight to balance the beam?
10. A uniform bridge span weighs 50 x 103 N and is 40.0 m long. An automobile weighing 15 x 103 N is
parked with its center of gravity located 12.0 m from the right pier. What upward support force is
provided by the left pier?
11. A child wants to use a 10 kg board that is 3.5 m long as a seesaw. Since all her friends are busy, she
balances the board by putting the support 1-m away from her when she sits at one end. What is her
mass?
12. A uniform board weighs 500 N and is 10.0 m long. It overhangs a building roof, extending over the edge
of the roof by 2.5 m. A paint bucket filled with sand, weighing 150 N, is sitting at the end of the board.
How far out is it safe for the 700 N worker to walk on the board? Assuming he can stretch no more than
0.5 m, can he reach the bucket when he needs it?
13. A 1.4-kg rod is supported by a single rope at an angle of 34° over the rod to its connection point on the
wall. The rod is attached to the wall on the other side by a hinge. Assume that the rod is uniform.
(a) What is the tension in the rope? (b) What are the two components of the support force exerted by the
hinge?
14. Which is easiest to rotate—when most of the mass is close to the axis or far from the axis?
15. Consider balancing a hammer upright on the tip of your finger. If the head of the hammer is heavy and
the handle long, would it be easier to balance with the end of the handle on your fingertip so that the
head is at the top, or the other way around with the head at your fingertip and the end of the handle at the
top?
16. Consider a pair of meter sticks standing nearly upright against a wall. If you release them, they’ll rotate
to the floor in the same time. But what if one has a massive hunk of clay stuck to its top end? Will it
rotate to the floor in a longer or shorter time?
17. The Earth moves about the Sun in an elliptical orbit. As the Earth moves closer to the Sun, does the
Earth-Sun system's moment of inertia increase, decrease, or remain constant?
18. Two hoops or rings (I = MR2) are centered, lying on a phonograph record. The smaller one has a radius
of 0.05 m and the larger a radius of 0.1 m. Both have a mass of 3 kg. What is the total moment of inertia
as the record turns around? Ignore the mass of the record.
19. A majorette takes two batons and fastens them together in the middle at right angles to make an "x"
shape. Each baton was 0.8 m long and each ball on the end is 0.20 kg. (Ignore the mass of the rods.)
What is the moment of inertia if the arrangement is spun around an axis through the center perpendicular
to both rods?
20. A uniform 10-m-long , 50-N ladder rests against a smooth vertical wall. If the ladder is just on the verge
of slipping when the angle it makes with the ground is 50º, find the coefficient of static friction between
the ladder and ground?
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21. A woman who weighs 500 N is standing on a board that weighs 100 N. The board is supported at each
end, and the support force at the right end is 3 times bigger than the support force at the left end. If the
board is 8 m long, how far from the right end is the woman standing?
22. A uniform 40-N board supports 2 children weighing500 N and 350 N. The support is under the center of
gravity of the board, and the 500-N child is 1.50 m from the center. A) Determine the upward force, N,
exerted on the board from the support. B) Determine where the 350-N child should sit to balance the
system.
23. A uniform, horizontal, 300-N beam, 5.00-m long, is attached to a wall by a pin connection that allows
the beam to rotate. Its far end is supported by a cable that makes an angle of 53.0 with the horixontal. If
a 600-N person stands 1.50-m from the wall, find (a) the tension in the cable and (b) the force exerted on
the beam by the wall.
24. You've climbed up to the top of a 7.5 m high telephone pole. Just as you reach the top, the pole breaks at
the base. Are you better off letting go of the pole and falling straight down, or sitting on top of the pole
and falling down to the ground on a circular path? Or does it make no difference?
25. The total kinetic energy of a baseball thrown with a spinning motion is a function of which of the
following?
a. its linear velocity but not rotational velocity
b. its rotational velocity but not linear velocity
c. both linear and rotational velocities
d. neither linear nor rotational velocity
26. Our galaxy may have begun as a huge cloud of gas and particles. Suppose the original cloud was far
larger than the present size of the galaxy, was more or less spherical, and was rotating very much more
slowly than at present. Gravitation between particles would have pulled them closer. What would be the
role of angular momentum conservation on the galaxy’s shape and present rotational speed?
27. A broom balances at its center of gravity. If you saw the broom into 2 parts through the center of gravity
and then weigh each part on a scale, which part will weigh more?
28. A skater, starting a spin with their arms extended, quickly pulls her arms in close to the body. What
effect does this have on the skater? Why?
29. The Earth's gravity exerts no torque on a satellite orbiting the Earth in an elliptical orbit. Compare the
motion at the point nearest the Earth (perigee) to the motion at the point farthest from the Earth
(apogee). At the point closest to the Earth
a. the angular velocity will be greatest although the linear speed will be the same.
b. the speed will be greatest although the angular velocity will be the same.
c. the kinetic energy and angular momentum will both be greater.
d. none of the above.
30. A 40 kg boy is standing on the edge of a stationary 30 kg platform that is free to rotate. The boy tries to
walk around the platform in a counterclockwise direction. As he does
a. the platform doesn't rotate.
b. the platform rotates in a clockwise direction just fast enough so that the boy remains stationary
relative to the ground.
c. the platform rotates in a clockwise direction while the boy goes around in a counterclockwise
direction relative to the ground.
d. both go around with equal angular velocities but in opposite directions.
31. A figure skater on ice with arms extended, spins at a rate of 2.0 rev/s. After she draws her arms in, she
spins at 5 rev/s. By what factor does her moment of inertia change in the process?
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