High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Physics Textbook (Master) > Angular / Rotational Motion

Conservation of Angular Momentum

Angular MomentumPrevious Section
Static EquilibriumNext Section
Popular Courses
Find My Course
0:00 / 0:00

Conservation of Angular Momentum


As we discussed, torque and angular momentum are related through the following equation:

τ⃗=r⃗×F⃗=r⃗×dP⟶dt=dL⟶dt\vec\tau=\vec r\times \vec F=\vec r\times\frac{\stackrel{\longrightarrow}{dP}}{dt}=\frac{\stackrel{\longrightarrow}{dL}}{dt}

Wize Tip
If the net external torque about an axis acting on a rotating body is zero, the angular momentum about the same axis remains constant
Lf=Li\boxed{L_f=L_i}


Exam Tip
We can either write down the angular momentum as L=r⊥mvL=r_{\perp}mv or L=IωL=I\omega depending on the information given to us in the question.

Example: Angular Momentum Conservation


In an isolated system, the moment of inertia of a rotating object is halved. What happens to the angular velocity of the object?
A. It is quartered.
B. It is halved.
C. It remains the same.
D. It is doubled.
E.It is quadrupled.

Solution: In an isolated system, there is no net torque. If there is no net torque on the system, then the total angular momentum of the system remains the same.
L=Iω

Therefore, if the moment of inertia, I, is halved, then for the angular momentum, L, to remain constant, the angular velocity, ω, must be doubled.

You're sitting on a rotating stool, initially rotating at 3.0 radians per second. You're holding a 5.0 kg dumbbell very close to you, 0.05 meters away from your center of rotation. You then extend your arms 1.0 meters away from you with the dumbbell. The moment of inertia of you and the stool is 4.0 kg m2. Neglect the mass of your arms and any rotational friction in the seat. What is your new angular velocity?


Extra Practice
angular momentum and conservation of angular momentum
A satellite is rotating once per minute. It has a moment of inertia of 10,000kg⋅m2. Erin, an astronaut, extends the satellite's solar panels, increasing its moment of inertia to 30,000kg⋅m2. How quickly is the satellite now rotating?
Friction turntable
Practice: Friction Turntable
Three blocks are placed on a spinning turntable at increasing distances away from the center of the turntable. Recall that the tangential velocity increases towards the edge of the turntable. The coefficient of static friction for all three blocks is the same, and they have the same mass. As the velocity of the turntable increases, which block slips first? Why?
A kid with mass m and velocity v runs toward a playground merry-go-round, which is initially at rest, and jumps on at its edge. The kid and the merry-go-round (mass M, radius R, and I = cMR2) then spin together.
We use the expression I = cMR2 because the merry-go-round is a complicated shape and we can't call it just a cylinder or a disk; c is some number between 0 and 1.
a) If the kid's initial velocity is tangent to the circular merry-go-round, what is the angular velocity after jumping on? Write your answer in terms of the other variables (mass, radius, C, and the tangential velocity). Then input your answer using m=30kg;M=150kg;C=0.55;R=2.3m;v=2.5m/sm=30kg; M = 150 kg; C = 0.55; R = 2.3 m; v = 2.5 m/s.
A horizontal platform in a shape of a disk rotates on a frictionless bearing about a vertical axle through the center of the disk. The platform has a mass of 200 kg and radius of 3.0 meters. A student with mass 80 kg, walks from the edge towards the center of the platform.
a) Find the moment of inertia of the disk.
b) If the angular speed of the disk is 2 rad/s, when the student moves another meter towards the center, find their angular speed.
A professional diver bends to reduce her moment of inertia by a factor of 1.8 before diving into a pool. What happens to her angular velocity, compared to if she had not bent before diving?
A uniform, spherical cloud of gas and dust from Orion has mass 2.0×1030 kg2.0\times10^{30}~\rm kg, has a radius of 1.0×1013 m1.0\times10^{13}~\rm m, and is rotating with a period of 1.4×1061.4\times10^6 years. The cloud collapses to form a star 7.0×108 m7.0\times10^8~\rm min radius. Find the star's period of rotation. Enter your result in days. (Moment of inertia for a solid sphere about an axis is 25MR2\frac25MR^2).
(In case you did not know: like all stars, the sun does rotate! Our sun's rotation period is 24 days.)
Ballistic pendulum and momentum
A ballistic pendulum is used to measure the speed of a bullet. It consists of a wooden block of mass M = 10kg suspended from a vertical string. A bullet of mass m = 100g strikes and embeds itself in the block and the block swings upward through a vertical distance of 0.10m. What was the bullet's speed? Enter your answer in m/s.
Object strikes a rod
An object of mass m and speed v0 strikes the bottom of a rigid uniform rod of length l and mass M that is hanging by a frictionless pivot in a wall. The pivot is through the center of mass of the rod, and is fastened to a wall, so that the rod can rotate about its center. Immediately after striking the rod, the object continues forward with half its initial speed.
a) Find the angular speed immediately after the collision
b) For what value of v0 will the rod just reach a horizontal position (before falling back down again)?
Spinning with a weight
You're sitting on a rotating stool, initially rotating at 3.0 radians per second. You're holding a 5.0 kg dumbbell very close to you, 0.05 meters away from your center of rotation. You then extend your arms 1.0 meters away from you with the dumbbell. The moment of inertia of you and the stool is 4.0 kg m2. Neglect the mass of your arms and any rotational friction in the seat.
a) What is your new angular velocity?
b) How would this question be different if you held a 2.5 kg dumbbell in each hand, instead of a single 5.0 kg weight, and extended them the same length?
Friction turntable
Three blocks are placed on a spinning turntable at increasing distances away from the center of the turntable. Recall that the tangential velocity increases towards the edge of the turntable. The coefficient of static friction for all three blocks is the same, and they have the same mass. As the velocity of the turntable increases, which block slips first? Why?
A rod is 3m long and rotates at ωo=5 rad/s.\omega_o=5~\rm rad/s.It is heated and expands to 3.2m. What is its new ω?\omega?
A uniform, spherical cloud of gas and dust from Orion has mass 2.0×1030 kg2.0\times10^{30}~\rm kg, has radius 1.0×1013 m1.0\times10^{13}~\rm m, and is rotating with a period of 1.4×1061.4\times10^6, years. The cloud collapses to form a star 70×108 m70\times10^8~\rm min radius. Find the star's rotation period.
A professional diver bends to reduce her moment of inertia by 1.8 times the initial moment of inertia and then dives into a pool. What happens to her angular velocity? Does it increase or decrease and by how much?
A kid with mass m and velocity v runs toward a playground merry-go-round, which is initially at rest, and jumps on at its edge. The kid and the merry-go-round (mass M, radius R, and I = cMR2) then spin together with a constant angular velocity ωf.
If kid's initial velocity is tangent to the circular merry-go-round, what is ωf?
What is the energy loss?
A horizontal platform in a shape of a disk rotates on a frictionless bearing about a vertical axle through the center of the disk. The platform has a mass of 200 kg and radius of 3m. A student with mass 80 kg, walks from the edge towards the center of the platform.
a) Find the moment of inertia of the disk
b) If the angular speed of the disk is 2 rad/s, when the student gets to 1 m towards the center, find the angular speed
An object of mass m and speed v0 strikes a rigid uniform rod of length l and mass M that is hanging by a frictionless pivot from the ceiling. Immediately after striking the rod, the object continues forward with half its initial speed.
a) Find the angular speed immediately after the collision
b) For what value of v0 will the rod just touch the ceiling?
A ladybug is on a rotating disk of radius R, moving steadily and radially toward the center of the disk as the disk rotates with constant angular velocity ω\omega. Which statement(s) is correct about the motion of the lady bug?
a) Its acceleration vector is always toward the center.
b) Its velocity vector has only a tangential component.
angular momentum and conservation of angular momentum
A satellite is rotating once per minute. It has a moment of inertia of 10,000kg⋅m2. Erin, an astronaut, extends the satellite's solar panels, increasing its moment of inertia to 30,000kg⋅m2. How quickly is the satellite now rotating?