Angular Acceleration
A disk of radius R = 0.25 m and mass M = 100 kg is mounted on a nearly frictionless axle. The disk is initially at rest. A string is wrapped lightly around the disk, and you pull on the string with a constant force F = 100 N
What is the angular speed, ω_f, after a time interval △t = 2.0 s? Note: Moment of inertia of a disk I=(1/2)MR^2
16 rad/s
Starting from rest, a woman lifts a barbell of mass mm with a constant force of F Newtons through a distance of h meters, at which point she is still lifting, and the barbell has acquired a speed of vm/s.
Let Ewoman stand for the internal energy of the woman.
For the system consisting of the woman + barbell + Earth, which equation below is the correct application of the energy principle from the initial state with the barbell at rest to the final state with the barbell having moved upward a distance h and has a speed v ?
ΔEwoman +(.5mv^2 − 0)+(mgh − 0) = 0
In lab you try to measure your approximate moment of inertia by holding two barbells and initially holding them with your hands outstretched, standing on a turntable. You have your friend spin the turntable, and after it is spinning at a steady speed, you pull your arms in. Each barbell has mass m = 6.0 kg, and with each of your arms outstretched, the distance of each barbell from the axis of rotation is r = 0.5 m. Initially, when you were spinning with the barbell carrying arms outstretched, you were spinning at an angular speed of ωi 3.0 rad/s. Your body plus the turntable has a moment of inertia of Ibody = 2.0 kg m^ 2
What is the final angular speed of rotation ωf?
[Assume: The moment of inertia of your outstretched hands, and the platform that you are standing on is negligible. Further, when you draw in your barbells, the barbells are aligned with the axis of your body, and so they do not contribute to moment of inertia.]
7.50 rad/s
Consider a uniform disk of mass, M and radius R as shown. Forces F1⃗ and F2⃗ act at the points shown in the figure.
What is the angular acceleration of the disk about an axis passing through the center of mass parallel to the z-axis?
<0,0,2/MR * ((F1)/2 − F2)>
A block of mass, M1 = 7 kg that hangs freely from a non-stretchable string of negligible mass that runs over a pulley and is connected to another block of mass, M2 = 10.00 kg that also hangs vertically down. The pulley can be modeled as a disk of mass m = 5 kg and radius R = 0.5 m. The rope does NOT slide over the surface of the pulley. Magnitude of acceleration due to gravity is g=10m/s^2. The pulley can be modeled as a disk (I=1/2mR^2)
What is the magnitude of the tension forces in each string shown in the figure?
T1 = 80.77 N, T2 = 84.62 N
A device consists of eight balls, each of mass M = 6 kg, attached to the ends of low-mass spokes of length L = 1.2 m, so the radius of rotation of the balls is L/2. The device is mounted in the vertical plane, as shown in the figure. The axle is held up by supports that are not shown, and the device rotates on the nearly frictionless axle. A lump of clay with mass m = 3 kg strikes at the bottom ball at an angle θ = 25∘, and sticks to it. Just before the impact the clay has a speed v = 5 m/s, and the device is rotating counterclockwise with angular speed ω = 0.5 rad/s
What is the initial angular momentum of the system (device + clay)?
<0,0,16.78>kg−m^2/s
A rigid beam of mass mmm, length L, has two supports. One support is located at the beam's left end, while the other is a distance 3L/4 from the left edge. A block of mass MMM is situated such that its center of mass is located a distance L/2 from the beam's left end as shown in the figure below.
What is the ratio of the normal force exerted by support 1 to the normal force exerted by support 2 on the beam?
F1 / F2 = 1/2
A system consists of a two concentric disks - a smaller uniform disk of mass, mmm and radius, r and a larger uniform disk of mass, M, and radius, R. The smaller disk lies on top of the larger disk. Force F1 acts tangentially (i.e. perpendicular to the radius) on the outer rim of the smaller disk as shown. Force F2 acts tangentially on the outer rim of the larger disk as shown.
What is the angular acceleration of the system about the axis? Note: The moment of inertia of a disk is I=.5MR^2
<0,0, 2(F_2 *R - F_1 * r) / (MR^2 + mr^2) >
Two small balls each of mass m are connected by a massless rod of length L. At a particular instant they have the velocities shown in the accompanying diagram. We are looking down on the system which is sliding on an icy surface, the x-y plane.
What is the rotational angular momentum of the system problem?
Lrot,A = <0,0,−mvL>
Consider a uniform rod of mass, M = 4 kg and length L = 1.6 m as shown. The rod is pivoted at point A at the left end of the rod. Force of magnitude F1 = 50 N acts at the right end of the rod and the direction of the force makes θ1 = 15∘ with the rod as shown. Force of magnitude F2 = 80 N acts at distance d = 1.12 m from the left end of the rod and the direction of the force makes θ2 = 70∘ with the rod as shown. Acceleration due to gravity is <0,−10,0>m/s2
What is the angular acceleration of the rod about the pivot point? Note: The moment of inertia of a rod about it's center of mass I = 1/12ML^ 2. You may need to use the parallel axis theorem.
<0,0,−28.03>rad/s2