Chapter 11: Angular Momentum

Adria Updike

11 Chapter 11: Angular Momentum

Section 11.1: Rolling Without Slipping

When we say an object is rolling, we mean that the object is rolling without slipping. Picture a bicycle wheel. To roll without slipping is for each point of the wheel to contact the ground while that point is at rest relative to the ground (only static friction between the wheel and the ground). For the wheel to slip, the entire wheel would move relative to the ground (kinetic friction) and would no longer be rolling. For our calculations, something is either rolling (rotational kinetic energy, maybe translational as well) or slipping (translational only) but not both at once.

The mathematical condition for rolling without slipping is that $v_{cm} = r \omega$ . This is only true for the center of mass of the object if no slippage is occurring.

Rolling friction refers to something that impedes the normal rotational motion of a rolling object. This might be something like sand, or a rubber roadway, or water, but something long its path deforms the shape of the wheel and leads to instabilities.

Section 11.2: Angular Momentum

Textbook Section 11.2: Angular Momentum

We discussed linear momentum in one and two dimensions in Chapter 8. Angular momentum is similar to linear momentum, but involves objects moving in a circular path instead of a straight line. An object can have both linear and angular momentum depending on how it is spinning. The angular momentum is denoted $\vec{L}$ , and has units of kg m $^2$ /s. The angular momentum is defined as

$\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times m \vec{v}$

$\vec{L}$ is defined in terms of the cross produce of the position vector $r$ of an object and the linear momentum $\vec{p} = m \vec{v}$ . The angular momentum is a vector quantity as a result. Just as force was the rate of change of linear momentum, torque is the rate of change of angular momentum.

$\frac{d \vec{L}}{dt} = \vec{r} \times \frac{d \vec{p}}{dt} = \vec{r} \times \vec{F} = \vec{\tau}$

We can also define $\vec{L}$ in terms of the torque equations.

$L = r m v \,\,sin\phi$

$\vec{L} = \vec{I} \omega$

Writing the angular momentum in terms of the moment of inertia means that we can now define the angular momentum for an extended object, any rigid body (not deforming as it moves). The direction of the angular momentum is found using the $\emph{right hand rule}$ . Point the fingers of your right hand in the direction of $\vec{r}$ , then curl them in the direction of $\vec{v}$ . Your thumb now points in the direction of the angular momentum vector $\vec{L}$ .

Section 11.3: Conservation of Angular Momentum

Textbook Section 11.3: Conservation of Angular Momentum

Like linear momentum, angular momentum is a conserved quantity as long as no external torques are applied while the mass and/or distances are changing. We can write the conservation of angular momentum as

$I_i \omega_i = I_f \omega_f.$

In Class Group Problem 11.1:

Calculate the angular momentum of the Sun. $M_{\odot} = 2 \times 10^{30}$ kg and $R_{\odot} = 6.95 \times 10^8$ m. The Sun completes one full rotation on its axis every 30 days on average.

When the Sun begins to die in about 5 billion years, it will expanding into a red giant star with a radius that extends out to the orbit of the Earth, 1.50 $\times 10^{11}$ m (without changing its mass). What will its angular velocity be at this point?

In Class Group Problem 11.3:

An alien 3.8 m tall throws a ball of clay at a door hinged from above (picture a really big doggie door). The 4.5 m tall door is initially hanging without rotating until the clay hits it. The clay sticks to the door and causes it to swing upwards in the opposite direction. How far off the ground does the door reach if it was just barely touching the ground below at rest? Solve this using the steps given below.

(a) The clay has a mass of 2.0 kg. The alien has a weight of 530 N and recoils 0.50 cm after throwing the class before coming to a rest. If the coefficient of kinetic friction between the alien and the ground is 0.900, what was the initial velocity of the clay?

(b) The clay is thrown horizontally, beginning 3.8 m above the ground. The door is 4.5 m tall. If the door is 6.0 m away from the alien, how far down the door from the hinge does the clay hit the door?

(c) Find the angular velocity $\omega$ of the door ( $m_{door}$ = 10 kg) if the clay sticks to the door and maintains a ball-like shape (treat the clay as a point particle).

$I_{door}$ = (1/3) M L $^2$

$I_{point \, particle}$ = M R $^2$

(d) What is the $y$ center of mass coordinate of the door-clay system?

(e) How high does the center of mass of the door rise above the ground before it falls back down again?

In Class Group Problem 11.4:

You have been asked to help evaluate a device to determine the speed of hockey pucks shot along the ice. The device consists of a rod which rests on the ice and is fastened at one end so that it is free to rotate horizontally. The free end of the rod has a small, light basket which will catch the hockey puck. The puck slides across the ice perpendicular to the rod and is caught in the basket which is initially at rest. The rod then rotates. Your friend who built this design plans to sell it to the local hockey league, but needs your help to figure out the speed of the puck as it hits the basket. The rod is 5 feet from the fixed location to the basket at the end. The rod has a mass of 7.0 kg, the basket has a mass of 0.40 kg, and the puck has a mass of 0.170 kg. Only the basket and puck are in contact with the ice, not the rod. To determine the coefficient of kinetic friction between the basket with puck and the ice, you give the stationary basket containing the puck a 10 N push until it’s moved 20 cm and watch it slide 35 feet across the ice before coming to a stop. When the puck hits the basket attached to the rod, it will rotate through $\theta$ degrees before coming to a stop due to friction between the basket and the puck. Design a formula for your friend to use to figure out the initial speed of the puck if he can measure how many degrees $\theta$ the rod rotates before stopping.

In Class Group Problem 11.5:

You are a member of a group designing an air filtration system for allergy suffers. To optimize its operation, you need to measure the mass of the common pollen in the air where the filter will be used. To measure the mass of the pollen, you have designed a small rectangular box with a hole in one side to allow the pollen to enter. Once the pollen is inside the box, it is given a positive electric charge and accelerated by an electrostatic force to a speed of 1.4 m/s. The pollen then hits the end of a very small, uniform bar which is hanging straight down from a pivot at its top. Since the bar has a negative charge at the tip, the pollen sticks to the bar as the bar swings up. Measuring the angle that the bar swings up would give the particle’s mass. After the angle is measured, the charge of the bar is reversed, releasing the particle. Your boss is skeptical that this idea will work, and tells you to calculate how small the mass of the bar needs to be to achieve an angle of 10 $^{\circ}$ from the vertical if a pollen particle has a mass of 4 $\times 10^{=9}$ g. Assume the pollen particle lands on the end of the bar and acts as a point particle. You don’t need to know anything about electric forces to solve this problem.

In Class Group Problem 11.6:

You have been asked to design a new stunt for the opening of an ice show. A small, 50 kg skater glides down a ramp and along a short level stretch of ice. While gliding along the level stretch, she makes herself as small as possible. Keeping herself as small as possible, she then grabs the bottom end of a large 180 kg vertical rod which is free to turn vertically about an axis through its center. The plan is for her to hold onto the 20 foot long rod while it swings her to the top. In order for the rod to swing all the way to the top of it’s trajectory while carrying her, the ramp she glides down must have some minimum initial height. Find the minimum height from which she must begin the stunt so that the stunt is successful if the rod has a uniform mass distribution and a moment of inertia of 1/3 M $r^2$ .

Practice Exam Question 11.1:

These questions are provided for practice purposes. There is no guarantee a problem similar to this one will be on your exam.

You have just been hired by United Airlines to save them money. It costs the airline about $0.59 to fly every pound they load onto the plane to its destination, so UA has decided to charge passengers by the pound (of themselves and their luggage) instead of charging a flat fee for a seat. However, they know that people lie about their weight, and they can’t ask people to step on a scale at the airport because that would cause too much customer backlash. Instead, they hired you with your strong physics background to design a sneaky scale for them to use instead. You design a rotating platform (large disk) and tell the passengers that they need to place their luggage on the disk and stand on it as well in order to scan them for dangerous materials not allowed on the plane. The disk has a mass of 350 lbs and a radius of 5.0 feet. The platform starts to rotate from rest when a torque of 92.3 Nm is applied to the edge for 6.0 seconds and then rotates at a constant angular velocity without friction. A passenger places himself and his bag 4.0 ft from the center of the platform as it rotates to be “scanned”. The disk slows down to an angular velocity of 1.70 rad/s as a result of the person and their luggage stepping on to the rotating platform. If UA decides to charge $3.00 per pound of person and luggage, how much is that person going to pay for their flight?

Hint: treat the person and their bag as point particles.

Practice Exam Question 11.2: