1. Physics 6A Rotational Motion Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

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

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

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

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

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

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

8. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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 = rθThis translates between distance (linear) and angle (rotational)

9. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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 = rθThis translates between distance (linear) and angle (rotational) Here are the other variables: v = rωlinear velocity relates to angular velocity a tan = rαlinear acceleration relates to angular acceleration Notice a pattern here?

10. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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 = rθThis translates between distance (linear) and angle (rotational) Here are the other variables: v = rωlinear velocity relates to angular velocity a tan = rαlinear acceleration relates to angular acceleration Multiply the angular quantity by the radius to get the corresponding linear quantity.

11. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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=x 0 +v 0 t+½at 2 θ=θ 0 +ω 0 t+½αt 2 v=v 0 +atω=ω0+αtω=ω0+αt v 2 =v 0 2 +2a(x-x 0 )ω 2 =ω 0 2 +2α(θ-θ 0 )

12. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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=x 0 +v 0 t+½at 2 θ=θ 0 +ω 0 t+½αt 2 v=v 0 +atω=ω0+αtω=ω0+αt v 2 =v 0 2 +2a(x-x 0 )ω 2 =ω 0 2 +2α(θ-θ 0 ) 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.

13. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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=x 0 +v 0 t+½at 2 θ=θ 0 +ω 0 t+½αt 2 v=v 0 +atω=ω0+αtω=ω0+αt v 2 =v 0 2 +2a(x-x 0 )ω 2 =ω 0 2 +2α(θ-θ 0 ) 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.

14. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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=x 0 +v 0 t+½at 2 θ=θ 0 +ω 0 t+½αt 2 v=v 0 +atω=ω0+αtω=ω0+αt v 2 =v 0 2 +2a(x-x 0 )ω 2 =ω 0 2 +2α(θ-θ 0 ) 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:

15. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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=x 0 +v 0 t+½at 2 θ=θ 0 +ω 0 t+½αt 2 v=v 0 +atω=ω0+αtω=ω0+αt v 2 =v 0 2 +2a(x-x 0 )ω 2 =ω 0 2 +2α(θ-θ 0 ) 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:

16. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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=x 0 +v 0 t+½at 2 θ=θ 0 +ω 0 t+½αt 2 v=v 0 +atω=ω0+αtω=ω0+αt v 2 =v 0 2 +2a(x-x 0 )ω 2 =ω 0 2 +2α(θ-θ 0 ) 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: Use a kinematics equation:

17. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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=x 0 +v 0 t+½at 2 θ=θ 0 +ω 0 t+½αt 2 v=v 0 +atω=ω0+αtω=ω0+αt v 2 =v 0 2 +2a(x-x 0 )ω 2 =ω 0 2 +2α(θ-θ 0 ) 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: Use a kinematics equation: Convert to revolutions:

18. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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=x 0 +v 0 t+½at 2 θ=θ 0 +ω 0 t+½αt 2 v=v 0 +atω=ω0+αtω=ω0+αt v 2 =v 0 2 +2a(x-x 0 )ω 2 =ω 0 2 +2α(θ-θ 0 ) 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: Use a kinematics equation: Convert to revolutions: (we did some rounding off)

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

20. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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. Page 306 in 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. You will not have to derive them, just know how to use them. 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:

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

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

23. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB h θ 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. 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?

24. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB h θ 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. 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?

25. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB h θ 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. we can replace ω with v/r so everything is in terms of the desired unknown 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?

26. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB h θ 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. we can replace ω with v/r so everything is in terms of the desired unknown 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?

27. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB h θ 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. 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 on page 279) 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?

28. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB h θ 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. 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 on page 279) Solid Sphere 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?

29. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB h θ 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. 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 on page 279) Solid SphereHollow Sphere 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?

30. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB h θ 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. 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 on page 279) Solid SphereHollow Sphere The solid sphere is faster because its moment of inertia is smaller. It reaches the bottom first. 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?