AP Physics – Unit 7 Rotational Motion Notes
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Rotational Motion Learning Objectives
3.F.1 Only the force component perpendicular to the line connecting the axis of rotation and the point
of application of the force results in a torque about that axis.
a. The lever arm is the perpendicular distance from the axis of rotation or revolution to the line
of application of the force.
b. The magnitude of the torque is the product of the magnitude of the lever arm and the
magnitude of the force.
c. The net torque on a balanced system is zero.
3.F.2 The presence of a net torque along any axis will cause a rigid system to change its rotational
motion or an object to change its rotational motion about that axis.
a. Rotational motion can be described in terms of angular displacement, angular velocity, and
angular acceleration about a fixed axis.
b. Rotational motion of a point can be related to linear motion of the point using the distance of
the point from the axis of rotation.
c. The angular acceleration of an object or rigid system can be calculated from the net torque
and the rotational inertia of the object or rigid system.
3.F.3 A torque exerted on an object can change the angular momentum of an object.
a. Angular momentum is a vector quantity, with its direction determined by a right-hand rule.
b. The magnitude of angular momentum of a point object about an axis can be calculated by
multiplying the perpendicular distance from the axis of rotation to the line of motion by the
magnitude of linear momentum.
c. The magnitude of angular momentum of an extended object can also be found by multiplying
the rotational inertia by the angular velocity.
d. The change in angular momentum of an object is given by the product of the average torque
and the time the torque is exerted.
4.D.2 The angular momentum of a system may change due to interactions with other objects or
systems.
a. The angular momentum of a system with respect to an axis of rotation is the sum of the
angular momenta, with respect to that axis, of the objects that make up the system.
b. The angular momentum of an object about a fixed axis can be found by multiplying the
momentum of the particle by the perpendicular distance from the axis to the line of motion of
the object.
c. Alternatively, the angular momentum of a system can be found from the product of the
system’s rotational inertia and its angular velocity.
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4.D.3 The change in angular momentum is given by the product of the average torque and the time
interval during which the torque is exerted.
5.A.2 For all systems under all circumstances, energy, charge, linear momentum, and angular
momentum are conserved. For an isolated or a closed system, conserved quantities are constant. An
open system is one that exchanges any conserved quantity with its surroundings.
5.E.1 If the net external torque exerted on the system is zero, the angular momentum of the system
does not change.
5.E.2 The angular momentum of a system is determined by the locations and velocities of the objects
that make up the system. The rotational inertia of an object or system depends upon the distribution of
mass within the object or system. Changes in the radius of a system or in the distribution of mass within
the system result in changes in the system’s rotational inertia, and hence in its angular velocity and
linear speed for a given angular momentum.
Note: Examples should include elliptical orbits in an Earth-satellite system.
Mathematical expressions for the moments of inertia will be provided where needed.
Students will not be expected to know the parallel axis theorem.
4.D.1 Torque, angular velocity, angular acceleration, and angular momentum are vectors and
can be characterized as positive or negative depending upon whether they give rise to or correspond to
counterclockwise or clockwise rotation with respect to an axis.
Angular Quantities
Rotational motion – motion where all points in an object move in circles
Axis of rotation – represents the “center” of all of the circles for an object in rotational motion
Radian - a way to measure an angle. To calculate an angle in radians, Θ = l/r where l is the arc length
and r is the radius of the circle. 1 revolution = 360o = 2π radians
Concept
Displacement
Velocity
Acceleration
Linear
x (m)
v (m/s)
a (m/s2)
Rotational
Θ (radians) theta
ω (radians/s) omega
α (radians/s2) alpha
Association
x=rΘ
v=rω
α=ra
Example 1 – A bike wheel rotates 4.5 revolutions. How many radians has it rotated?
If the radius of the bike wheel is .5 m, what linear distance does a point on the outside of the wheel
travel?
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Example 2 – What is the relationship between linear velocity and angular velocity?
What is the relationship between linear acceleration and angular acceleration?
Example 3 – On a rotating carousel, one child sits on a horse near the outer edge and another child sits
on a lion halfway out from the center:
a)
Which child has the greater linear velocity?
b) Which child has the greater angular velocity?
c) Which child has the greater centripetal acceleration?
Rotational Kinematics:
Linear
v = vo + at
Δx = vot + ½ at2
v2 = vo2 + 2a Δx
Rotational
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Example 4 – A child is on a carousel which is initially at rest. At time t=0 it is given a constant angular
acceleration of .06 rad/s2, which increases its angular velocity for 8.0 s. The radius of the carousel is
2.5m At t=8.0s, determine:
a)
The angular velocity of the carousel
b) The linear (tangential) velocity of the child
c) The linear (tangential) acceleration of the child
d) The centripetal acceleration of the child
Rolling Motion (without slipping)
v=rω
Example 5 – A bicycle slows down uniformly from vo = 8.4 m/s to rest over a distance of 115m. Each
wheel has radius of .34m
a) Determine the angular velocity of the wheels at t=0s
b) Determine the total number of radians each wheel rotates before coming to rest
c) Determine the number of revolutions each wheel rotates before coming to rest
d) Determine the angular acceleration of the wheel
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Rotational Dynamics
Torque – a force which causes an object to rotate about an axis.
Moment arm (aka lever arm) – the perpendicular distance from the axis of rotation to the line about
which the force acts
Steps to solving torque problems:
a) Draw a clear and complete diagram
b) Choose the object or objects that will be the system to be studied.
c) Draw a free body diagram for the object showing all forces acting on it and where they act so
you can determine the torque due to each.
d) Identify the axis of rotation and determine the torques about it. Choose positive and negative
directions (clockwise and counterclockwise)
e) Apply Newton’s second law for rotation to get the net torque. If the object is not rotating, net
torque = 0 Nm, so you can set clockwise torques = counterclockwise torques. Tc = Tcc
Example 6 – What is the magnitude of the torque acting on the lever in each case?
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Example 7 – How far should the smaller person sit from the fulcrum to balance the seesaw?
Example 8 – Calculate the net torque acting on this wheel
Example 9 - A horizontal, uniform board of weight 100 N and length 8 m is supported by vertical chains at each
end. A person weighing 500 N is sitting on the board. The tension in the right chain is 250 N.
a) Draw a picture depicting the situation
b) What is the tension in the left chain?
c)
How far from the left end of the board is the person sitting?
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Rotational Dynamics - Torque and Rotational Inertia
Moment of Inertia (or rotational inertia) – a measure of how much an object resists rotation. An object
resists rotation based upon its shape (is it a ring? A disk? A sphere?), its mass, and its radius.
Linear
Force
Rotational equivalent
Mass
Acceleration
F=ma
or
ΣF = manet
τ=Iα
or
Σ τ = I αnet
Example 10 - The rotational inertia of a ring can be calculated using the formula I = mr2
The rotational inertia of a disk can be calculated using the formula I = ½ mr2
If they both have the same mass and the same radius, which has the greater rotational inertia, a ring or
a disk?
Which one (the disk or the ring) would be easier to rotate? Why?
Would the rotational inertia of a disk (or a ring) be affected if it were in space? Why or why not?
Example 11 - A wheel that has a moment of inertia of 50 kg-m2 is pulled on by two ropes as shown
below. One rope is wrapped around the inner radius of 0.1 m and the other is wrapped around the
outer radius of 0.4 m.
a) Determine the angular acceleration of the wheel.
b)
If the forces are applied for 10 seconds, what is the resulting angular velocity?
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Rotational Kinetic Energy
Linear
K = ½ mv2
W = Fd
Rotational
K = ½ Iω2
W = τθ
At any given point in time an object may possess any of the following:



Potential energy
Translational kinetic energy (1/2 mv2)
Rotational kinetic energy (1/2 Iω2 )
Example 12 – A disk of mass M and radius R sits on top of a ramp of height h.
a) What type of energy does it possess when it is at rest at the top of the ramp?
b) It begins rolling down the ramp without slipping. What type of energy does it possess at the
bottom of the ramp? Be specific.
c) Write an equation showing conservation of energy where “before” is the top of the ramp, and
“after” is the bottom of the ramp. Moment of inertia for a disk is I=1/2 mr2
d) Suppose the disk slides down the ramp without friction and thus without rolling. Write an
equation showing conservation of energy between the top and bottom of the ramp.
e) In which scenario, c or d, will the disk have the greater speed at the bottom of the ramp?
Explain.
Example 13 – A wheel is spinning in place (not rolling). A torque of 15 Nm is applied and the wheel stops
spinning after 3 revolutions. How much work was done to stop the wheel?
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Angular Momentum
Linear
Rotational
p = mv
L = Iω
Δp = FΔt
ΔL = τΔt
If an object is rotating, it has angular momentum. Note the relationships above between the linear
momentum formulas and the rotational momentum formulas.
Law of Conservation of Angular Momentum – The total angular momentum of a rotating object
remains constant if the net torque acting on it is zero.
Example 14 –
a) What is the angular momentum of a 3 kg ring of radius 2m rotating at an angular speed of 12
rad/s? I=mr2 for a ring.
b) How much torque is required to stop the ring in 6 seconds?
c) What is the angular acceleration during this time period?
d) What angular distance (in radians) is covered during this time period?
Example 15 – A student spins on a stool with their arms outstretched. At this point, they have an
angular momentum of value L.
a)
What happens to the students angular velocity as they draw their arms in closer to their body?
b) What happens to the angular momentum of the student after they have drawn in their arms?
c) Given the formula for angular momentum, justify what must have occurred.
AP Physics – Unit 7 Rotational Motion Notes
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