Download Torque and Rotational Motion

Torque and Rotational
Motion
AP Physics 1
Angular Kinematics
Discussion Questions
• How do we get around?
• How do we move things?
Angular Displacement
• We’ve discussed linear motion, but what happens when things roll or
rotate?
• If we look at a wheel, how far will it travel linearly in one rotation?
• Imagine we took the tire off the wheel and stretched it out long!
Angular Displacement
• When referring to how many times something rotates, we call it
angular displacement or the change in the angle theta
• Angular displacement, θ
• Measured in radians
• How do we convert from radians to meters?
• 𝑥 = 𝑟∆𝜃
Δθ
r
Angular Displacement
• A disk rotates 5 times. How many radians is this?
2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
5 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 ×
= 10 π 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ≈ 30 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛
• If the disk has a radius of .3 m, how many meters has it rotated
through?
𝑥 = 𝑟∆𝜃
𝑥 = 0.3𝑚 ∗ 10𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 3𝜋 𝑚 ≈ 9 𝑚
OR
2𝜋 (.3 𝑚)
5 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 ×
= 3π 𝑚 ≈ 9 𝑚
1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛
Angular Velocity
• Just like linear velocity is the rate of change of displacement, angular
velocity it the rate of change of angular displacement.
∆𝜃
𝜔=
∆𝑡
• Angular velocity is represented by the lower case Greek letter omega
and is measured in radians per second.
Angular Velocity
• We know a relationship exists between linear displacement and angular displacement, so
it there a relationship between linear velocity and angular velocity?
YES!
• If a wheel with a radius of 0.5 m travels 3 meters in 1 second we know it’s linear velocity
is 3 m/s. We also know that the linear distance traveled is directly related to the
circumference of the wheel.
3𝑚
1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛
1 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
∗
∗
∗
= 6 𝑟𝑎𝑑/𝑠𝑒𝑐
𝑠 1 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒
2𝜋 .5 𝑚
1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛
𝑣 = 𝑟𝜔
Angular Velocity
• A disk rotates 5 times in 2 seconds. What is it’s angular velocity?
2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
5 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 ×
= 10 π 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ≈ 30 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛
∆𝜃 10 π 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
𝑟𝑎𝑑
𝜔=
=
=5𝜋
∆𝑡
2 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
𝑠𝑒𝑐
• What is this in meters per second if the disk has a radius of 0.3 m?
𝑟𝑎𝑑
𝑚
𝑣 = 𝑟𝜔 = 0.3𝑚 ∗ 5 𝜋
= 1.5𝜋
𝑠𝑒𝑐
𝑠
• This assumes that the disk is rotating at a constant rate, but what happens
if it is not rotating at a constant angular velocity?
Angular Acceleration
• Angular acceleration is the rate of change in angular velocity with
respect to time just like linear acceleration is the rate of change in
linear velocity with respect to time.
∆𝜔
𝛼=
∆𝑡
• It is represented by the lower case Greek letter alpha and is measured
in radians per second squared.
Angular Acceleration
• Just as with the previous two quantities, we can easily convert
between angular acceleration and linear acceleration. It even follows
the same relationship!
𝑎 = 𝑟𝛼
Angular Acceleration
• When there is angular acceleration, the angular displacement and
angular velocity is effected in the same way that linear acceleration
effects linear displacement and linear velocity.
1 2
𝜃 = 𝜃0 + 𝜔0 𝑡 + 𝛼𝑡
2
𝜔 = 𝜔0 + 𝛼𝑡
𝜔2 = 𝜔02 + 𝛼𝑡
Angular Acceleration
A disk with a radius of 0.2 m starts at rests and uniformly accelerates to
10 rad/sec in 4 seconds.
• What is the disk’s angular acceleration? 𝛼 =
∆𝜔
∆𝑡
=
10−0
4
𝑟𝑎𝑑
𝑠𝑒𝑐 2
𝑚
0.5 2
𝑠
= 2.5
• What is the disk’s linear acceleration? 𝑎 = 𝑟𝛼 = .2 ∗ 2.5 =
• How many times has the disk rotated in 4 seconds?
1 2
1
𝜃 = 𝜃0 + 𝜔0 𝑡 + 𝛼𝑡 = 0 + 0 + 2.5 42 = 20 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
2
2
1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 10
20 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ∗
=
𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠
2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
𝜋
Comparison of linear and angular quantities
Linear Quantity
Angular Quantity
Displacement, x
Angular Displacement, θ
∆𝑥
Velocity, 𝑣𝑥 =
∆𝑡
∆𝑣𝑥
Acceleration, 𝑎𝑥 =
∆𝑡
∆𝜃
Angular Velocity, 𝜔 =
∆𝑡
∆𝜔
Angular Acceleration, 𝛼 =
∆𝑡
Converting between linear and angular quantities
𝑥 = 𝑟∆𝜃
𝑣 = 𝑟𝜔
𝑎 = 𝑟𝛼
Comparison of linear and angular kinematic
formulas
Linear Quantity
Angular Quantity
1
𝑥 = 𝑥0 + 𝑣𝑥0 𝑡 + 𝑎𝑥 𝑡 2
2
1 2
𝜃 = 𝜃0 + 𝜔0 𝑡 + 𝛼𝑡
2
𝑣𝑥 = 𝑣𝑥0 + 𝑎𝑥 t
𝜔 = 𝜔0 + 𝛼𝑡
2
𝑣𝑥
=
2
𝑣𝑥0
+2𝑎𝑥 (𝑥 − 𝑥0 )
2
𝜔 =
2
𝜔0
+ 𝛼𝑡
Torque
Discussion Questions
• What causes things to rotate?
• How do you get things to rotate faster or slower?
Torque
•
•
•
•
•
•
•
•
•
What is a force?
A torque is a twist!
𝜏 = 𝑟 × 𝐹
r is the distance between the axis of rotation and the location of the force.
x is a cross product. A cross product is similar to the dot product that we saw in work,
except that now you take the sine instead of the cosine.
𝜏 = 𝑟⊥ 𝐹 = 𝑟𝐹 sin 𝜃
Torque is represented by the lower case Greek letter Tau and is measured in NewtonMeters
Torque is a vector quantity however the direction is not up, down, left, or right! It is
clockwise or counterclockwise since torques cause rotations. (Counterclockwise is
conventionally positive)
It is the rotational analogue of force.
Torque
• If the board to the right is
3 meters long and a force
of F=5N is applied at
r=2.8 m. What is the
torque on the board?
• 𝜏 = 2.8 × 5 =
14 𝑁𝑚 𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒
r
F
Torque
F
• If the board to the right is
3 meters long and a force
of F=5N is applied at
r=2.8 m. What is the
torque on the board?
• 𝜏 = 2.8 ∗ 5 sin 45° =
9.9 𝑁𝑚 𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒
r
45°
Torque
• What happens when multiple forces act
on an object?
FN
Fa
• What happens when multiple torques
act on an object?
• When dealing with multiple torques,
make sure it is clear where your axis of
rotation is since this is how r is
measured.
r
Ff
Fg
Equilibrium
• Is the box below in translational equilibrium?
• YES!
5N
• Is it in rotational equilibrium?
• NO! It will rotate clockwise.
5N
Torque and Static Equilibrium
• We know a system is in translational equilibrium when the sum of
forces is equal to zero.
• A system is in rotational equilibrium when the sum of the torques is
equal to zero.
• A system is in static equilibrium when both the sum of forces and the
sum of torques are equal to zero.
𝐹=0
𝜏=0
Torque and Static Equilibrium
• A child is sitting on a swing. If the child
has a mass of 30 kg, what must the forces
in the left and right chains be to hold up
the child. The swing seat is 0.5 meters
long and of negligible mass. Assume the
child’s weight is at the center mass of the
swing.
FL
FR
R
r
Fg
𝐹 = 𝐹𝐿 + 𝐹𝑅 − 𝐹𝑔 = 0
𝐹𝐿 + 𝐹𝑅 = 𝐹𝑔 = 𝑚𝑔 = 30 ∗ 10 = 300𝑁
𝜏𝑎𝑏𝑜𝑢𝑡 𝑙𝑒𝑓𝑡 𝑐ℎ𝑎𝑖𝑛 = 0 ∗ 𝐹𝐿 + 𝑅𝐹𝑅 − 𝑟𝐹𝑔 = 0
0 + 0.5𝐹𝑅 − 0.25 ∗ 300 = 0
𝐹𝑅 = 150𝑁
𝐹𝐿 + 150𝑁 = 300 𝑁
𝐹𝐿 = 150𝑁
Torque
• What happens when the system is not in static equilibrium?
• The same thing that happens for forces! 𝑎 =
𝜏
𝐹
𝑚
=
𝐹 𝑛𝑒𝑡
𝑚
𝜏𝑛𝑒𝑡
𝛼=
=
𝐼
𝐼
• Forces cause linear accelerations, torques cause angular
accelerations.
• The I in the above formula stands for the rotational inertia or moment
of inertia and is specific to the object and how the object is rotating.
Rotational Inertia or Moment of Inertia
• If you remember inertia is an object’s resistance to change. It is
directly proportional to mass.
• An object’s rotational inertia or moment of inertia is the object’s
resistance to rotations. These are impacted by the size, shape and axis
of rotation of the object.
• I=Ʃmr2 for systems of particles
• I=kmr2 for continuous objects, where k is a constant (usually a
fraction)
• You will NOT need to memorize the moment of inertia for specific
objects.
Common Moments of Inertia
http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html
Torque
• What is the angular acceleration of a 2 kg disk that has a 5Nm torque
applied to it at it’s radius of 0.1m? Idisk= ½ mr2
•𝛼=
𝜏
𝐼
=
𝜏𝑛𝑒𝑡
𝐼
=1
2
5
∗2∗0.12
=
𝑟𝑎𝑑
500 2
𝑠
Angular Momentum
Angular Momentum
•
•
•
•
Angular momentum is the rotational counterpart of linear momentum.
𝑝 = 𝑚𝑣
𝐿 = 𝑟 × 𝑝 = 𝐼𝜔
There are 2 different representations of angular momentum. One involves
the cross product of distance between the axis of rotation and location of
object and linear momentum and the other the moment of inertia, I, and
angular velocity, ω.
• Just as there is a relationship between linear momentum and force, ∆𝑝 =
𝐹∆𝑡, there is also a relationship between angular momentum and
torque,∆𝐿 = 𝜏∆𝑡.
Conservation of Angular Momentum
If no net external torques act on a system then the system’s angular
momentum, L, remains constant.
Angular Momentum
An ice skater begins a spin with his arms out. His angular velocity at
the beginning of the spin is 2.0 rad/s and his moment of inertia is 6
kgm2. As the spin proceeds he pulls in his arms decreasing his
moment of inertia to 4.5 kgm2. What is the angular velocity after
pulling in his arms?
Lo  L
Io  I
(6)( 2)  (4.5)
  2.67 rad/s
AP Physics C- Rotational Motion- Part II; Dolores Gende