1. Physics 2 Rotational Motion Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. First some quick geometry review:
We need this formula for arc length to see the connection between rotational motion and linear motion.
x = rθ θ
We will also need to be able to convert from revolutions to radians.
There are 2π radians in one complete revolution. r
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. Angular Acceleration = α =
Definitions of angular velocity and angular acceleration are analogous to what we had for linear motion.
x = rθ
This is the Greek letter omega (not w) θ
Angular Velocity = ω = r
Angular Acceleration = α =
This is the Greek letter alpha (looks kinda like a fish)
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

4. Angular Acceleration = α =
Definitions of angular velocity and angular acceleration are analogous to what we had for linear motion.
x = rθ
This is the Greek letter omega (not w) θ
Angular Velocity = ω = r
Angular Acceleration = α =
This is the Greek letter alpha (looks kinda like a fish)
Example: A centrifuge starts from rest and spins for 7 seconds until it reaches 1000 rpm.
Find the final angular velocity and the angular acceleration (assume constant).
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

5. Angular Acceleration = α =
Definitions of angular velocity and angular acceleration are analogous to what we had for linear motion.
x = rθ
This is the Greek letter omega (not w) θ
Angular Velocity = ω = r
Angular Acceleration = α =
This is the Greek letter alpha (looks kinda like a fish)
Example: A centrifuge starts from rest and spins for 7 seconds until it reaches 1000 rpm.
Find the final angular velocity and the angular acceleration (assume constant).
rpm stands for “revolutions per minute” – we can treat the first part of this problem just like a unit conversion:
Standard units for angular velocity are radians per second
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

6. Angular Acceleration = α =
Definitions of angular velocity and angular acceleration are analogous to what we had for linear motion.
x = rθ
This is the Greek letter omega (not w) θ
Angular Velocity = ω = r
Angular Acceleration = α =
This is the Greek letter alpha (looks kinda like a fish)
Example: A centrifuge starts from rest and spins for 7 seconds until it reaches 1000 rpm.
Find the final angular velocity and the angular acceleration (assume constant).
rpm stands for “revolutions per minute” – we can treat the first part of this problem just like a unit conversion:
Standard units for angular velocity are radians per second
Now we can use the definition of angular acceleration:
Standard units for angular acceleration are radians per second squared.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

7. We already know how to deal with linear motion.
We have formulas for kinematics, forces, energy and momentum.
We will find similar formulas for rotational motion. Actually, we already know the formulas – they are the same as the linear ones we already know!
All you have to do is translate the variables.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

8. We already know how to deal with linear motion.
We have formulas for kinematics, forces, energy and momentum.
We will find similar formulas for rotational motion. Actually, we already know the formulas – they are the same as the linear ones we already know!
All you have to do is translate the variables.
We have already seen one case:
x = rθ This translates between distance (linear) and angle (rotational)
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

9. We already know how to deal with linear motion.
We have formulas for kinematics, forces, energy and momentum.
We will find similar formulas for rotational motion. Actually, we already know the formulas – they are the same as the linear ones we already know!
All you have to do is translate the variables.
We have already seen one case:
x = rθ This translates between distance (linear) and angle (rotational)
Here are the other variables:
v = rω linear velocity relates to angular velocity
atan = rα linear acceleration relates to angular acceleration
Notice a pattern here?
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

10. We already know how to deal with linear motion.
We have formulas for kinematics, forces, energy and momentum.
We will find similar formulas for rotational motion. Actually, we already know the formulas – they are the same as the linear ones we already know!
All you have to do is translate the variables.
We have already seen one case:
x = rθ This translates between distance (linear) and angle (rotational)
Here are the other variables:
v = rω linear velocity relates to angular velocity
atan = rα linear acceleration relates to angular acceleration
Multiply the angular quantity by the radius to get the corresponding linear quantity.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

11. Linear Motion (constant a) Rotational Motion (constant α)
We can use this technique to find angular motion formulas from the analogous linear motion formulas we already know. Here are some kinematics formulas.
Linear Motion (constant a)
Rotational Motion (constant α)
x=x0+v0t+½at2
θ=θ0+ω0t+½αt2
v=v0+at
ω=ω0+αt
v2=v02+2a(x-x0)
ω2=ω02+2α(θ-θ0)
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

12. Linear Motion (constant a) Rotational Motion (constant α)
We can use this technique to find angular motion formulas from the analogous linear motion formulas we already know. Here are some kinematics formulas.
Linear Motion (constant a)
Rotational Motion (constant α)
x=x0+v0t+½at2
θ=θ0+ω0t+½αt2
v=v0+at
ω=ω0+αt
v2=v02+2a(x-x0)
ω2=ω02+2α(θ-θ0)
Here is a kinematics example: A cyclist starts from rest and accelerates to a speed of 15 m/s in a time of 25 sec. Assume the tires have a diameter of 700mm and that the acceleration is constant.
Find the total number of revolutions that the wheels make during the 25 second interval.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

13. Linear Motion (constant a) Rotational Motion (constant α)
We can use this technique to find angular motion formulas from the analogous linear motion formulas we already know. Here are some kinematics formulas.
Linear Motion (constant a)
Rotational Motion (constant α)
x=x0+v0t+½at2
θ=θ0+ω0t+½αt2
v=v0+at
ω=ω0+αt
v2=v02+2a(x-x0)
ω2=ω02+2α(θ-θ0)
Here is a kinematics example: A cyclist starts from rest and accelerates to a speed of 15 m/s in a time of 25 sec. Assume the tires have a diameter of 700mm and that the acceleration is constant.
Find the total number of revolutions that the wheels make during the 25 second interval.
We basically have two options on how to proceed. We can switch to angular variables right away, or we can do the corresponding problem in linear variables and translate at the end.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

14. Linear Motion (constant a) Rotational Motion (constant α)
We can use this technique to find angular motion formulas from the analogous linear motion formulas we already know. Here are some kinematics formulas.
Linear Motion (constant a)
Rotational Motion (constant α)
x=x0+v0t+½at2
θ=θ0+ω0t+½αt2
v=v0+at
ω=ω0+αt
v2=v02+2a(x-x0)
ω2=ω02+2α(θ-θ0)
Here is a kinematics example: A cyclist starts from rest and accelerates to a speed of 15 m/s in a time of 25 sec. Assume the tires have a diameter of 700mm and that the acceleration is constant.
Find the total number of revolutions that the wheels make during the 25 second interval.
Switching to angular variables right away:
Convert to angular velocity:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

15. Linear Motion (constant a) Rotational Motion (constant α)
We can use this technique to find angular motion formulas from the analogous linear motion formulas we already know. Here are some kinematics formulas.
Linear Motion (constant a)
Rotational Motion (constant α)
x=x0+v0t+½at2
θ=θ0+ω0t+½αt2
v=v0+at
ω=ω0+αt
v2=v02+2a(x-x0)
ω2=ω02+2α(θ-θ0)
Here is a kinematics example: A cyclist starts from rest and accelerates to a speed of 15 m/s in a time of 25 sec. Assume the tires have a diameter of 700mm and that the acceleration is constant.
Find the total number of revolutions that the wheels make during the 25 second interval.
Switching to angular variables right away:
Convert to angular velocity:
Find angular acceleration:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

16. Linear Motion (constant a) Rotational Motion (constant α)
We can use this technique to find angular motion formulas from the analogous linear motion formulas we already know. Here are some kinematics formulas.
Linear Motion (constant a)
Rotational Motion (constant α)
x=x0+v0t+½at2
θ=θ0+ω0t+½αt2
v=v0+at
ω=ω0+αt
v2=v02+2a(x-x0)
ω2=ω02+2α(θ-θ0)
Here is a kinematics example: A cyclist starts from rest and accelerates to a speed of 15 m/s in a time of 25 sec. Assume the tires have a diameter of 700mm and that the acceleration is constant.
Find the total number of revolutions that the wheels make during the 25 second interval.
Switching to angular variables right away:
Convert to angular velocity:
Use a kinematics equation:
Find angular acceleration:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

17. Linear Motion (constant a) Rotational Motion (constant α)
We can use this technique to find angular motion formulas from the analogous linear motion formulas we already know. Here are some kinematics formulas.
Linear Motion (constant a)
Rotational Motion (constant α)
x=x0+v0t+½at2
θ=θ0+ω0t+½αt2
v=v0+at
ω=ω0+αt
v2=v02+2a(x-x0)
ω2=ω02+2α(θ-θ0)
Here is a kinematics example: A cyclist starts from rest and accelerates to a speed of 15 m/s in a time of 25 sec. Assume the tires have a diameter of 700mm and that the acceleration is constant.
Find the total number of revolutions that the wheels make during the 25 second interval.
Switching to angular variables right away:
Convert to angular velocity:
Use a kinematics equation:
Find angular acceleration:
Convert to revolutions:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

18. Linear Motion (constant a) Rotational Motion (constant α)
We can use this technique to find angular motion formulas from the analogous linear motion formulas we already know. Here are some kinematics formulas.
Linear Motion (constant a)
Rotational Motion (constant α)
x=x0+v0t+½at2
θ=θ0+ω0t+½αt2
v=v0+at
ω=ω0+αt
v2=v02+2a(x-x0)
ω2=ω02+2α(θ-θ0)
Here is a kinematics example: A cyclist starts from rest and accelerates to a speed of 15 m/s in a time of 25 sec. Assume the tires have a diameter of 700mm and that the acceleration is constant.
Find the total number of revolutions that the wheels make during the 25 second interval.
This time do the linear problem first:
Find linear acceleration:
Convert to revolutions:
Use a kinematics equation:
(we did some rounding off)
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

19. All of these have analogues for rotational motion as well.
Some other major topics for linear motion are Energy, Forces and Momentum.
All of these have analogues for rotational motion as well.
Any moving object will have Kinetic Energy. This applies to rotating objects. Here’s a Formula:
We know that ω is angular velocity. Comparing with the formula for linear kinetic energy, what do you think I is?
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

20. All of these have analogues for rotational motion as well.
Some other major topics for linear motion are Energy, Forces and Momentum.
All of these have analogues for rotational motion as well.
Any moving object will have Kinetic Energy. This applies to rotating objects. Here’s a Formula:
We know that ω is angular velocity. Comparing with the formula for linear kinetic energy, what do you think I is?
The I in our formula takes the place of m (mass) in the linear formula. We call it Moment of Inertia (or rotational inertia). It plays the same role in rotational motion that mass plays in linear motion (I quantifies how difficult it is to produce an angular acceleration, just like mass relates to linear acceleration).
The value for I will depend on the shape of your object, but the basic rule of thumb is that the farther the mass is from the axis of rotation, the larger the inertia.
Your book has a table of formulas for different shapes.
These are all based on the formula for the moment of inertia of a point particle.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

21. Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

22. Example
A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle θ above the horizontal. Both spheres start from rest at the same vertical height h.
a) How fast is each sphere moving when it reaches the bottom of the hill?
b) Which sphere will reach the bottom first, the hollow one or the solid one?
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

23. Example
A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle θ above the horizontal. Both spheres start from rest at the same vertical height h.
a) How fast is each sphere moving when it reaches the bottom of the hill?
b) Which sphere will reach the bottom first, the hollow one or the solid one?
We can use conservation of energy for this one. Since they don’t mention it we can ignore rolling friction and just assume that the total energy at the top equals the energy at the bottom. h θ
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

24. Example
A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle θ above the horizontal. Both spheres start from rest at the same vertical height h.
a) How fast is each sphere moving when it reaches the bottom of the hill?
b) Which sphere will reach the bottom first, the hollow one or the solid one?
We can use conservation of energy for this one. Since they don’t mention it we can ignore rolling friction and just assume that the total energy at the top equals the energy at the bottom. h θ
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

25. Example
A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle θ above the horizontal. Both spheres start from rest at the same vertical height h.
a) How fast is each sphere moving when it reaches the bottom of the hill?
b) Which sphere will reach the bottom first, the hollow one or the solid one?
We can use conservation of energy for this one. Since they don’t mention it we can ignore rolling friction and just assume that the total energy at the top equals the energy at the bottom. h θ
we can replace ω with v/r so everything is in terms of the desired unknown
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

26. Example
A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle θ above the horizontal. Both spheres start from rest at the same vertical height h.
a) How fast is each sphere moving when it reaches the bottom of the hill?
b) Which sphere will reach the bottom first, the hollow one or the solid one?
We can use conservation of energy for this one. Since they don’t mention it we can ignore rolling friction and just assume that the total energy at the top equals the energy at the bottom. h θ
we can replace ω with v/r so everything is in terms of the desired unknown
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

27. Example
A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle θ above the horizontal. Both spheres start from rest at the same vertical height h.
a) How fast is each sphere moving when it reaches the bottom of the hill?
b) Which sphere will reach the bottom first, the hollow one or the solid one?
We can use conservation of energy for this one. Since they don’t mention it we can ignore rolling friction and just assume that the total energy at the top equals the energy at the bottom. h θ
we can replace ω with v/r so everything is in terms of the desired unknown
At this point we can substitute the formula for each shape (from table 9.2)
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

28. Example
A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle θ above the horizontal. Both spheres start from rest at the same vertical height h.
a) How fast is each sphere moving when it reaches the bottom of the hill?
b) Which sphere will reach the bottom first, the hollow one or the solid one?
We can use conservation of energy for this one. Since they don’t mention it we can ignore rolling friction and just assume that the total energy at the top equals the energy at the bottom. h θ
we can replace ω with v/r so everything is in terms of the desired unknown
At this point we can substitute the formula for each shape (from table 9.2)
Solid Sphere
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

29. Example
A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle θ above the horizontal. Both spheres start from rest at the same vertical height h.
a) How fast is each sphere moving when it reaches the bottom of the hill?
b) Which sphere will reach the bottom first, the hollow one or the solid one?
We can use conservation of energy for this one. Since they don’t mention it we can ignore rolling friction and just assume that the total energy at the top equals the energy at the bottom. h θ
we can replace ω with v/r so everything is in terms of the desired unknown
At this point we can substitute the formula for each shape (from table 9.2)
Solid Sphere
Hollow Sphere
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

30. The solid sphere is faster because its moment of inertia is smaller.
Example
A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle θ above the horizontal. Both spheres start from rest at the same vertical height h.
a) How fast is each sphere moving when it reaches the bottom of the hill?
b) Which sphere will reach the bottom first, the hollow one or the solid one?
We can use conservation of energy for this one. Since they don’t mention it we can ignore rolling friction and just assume that the total energy at the top equals the energy at the bottom. h θ
we can replace ω with v/r so everything is in terms of the desired unknown
At this point we can substitute the formula for each shape (from table 9.2)
Solid Sphere
Hollow Sphere
The solid sphere is faster because its moment of inertia is smaller.
It reaches the bottom first.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB