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Physics 104

Review - Angular Motion


Write an expression for (a) the distance x moved by a particle traveling with a constant linear velocity v in time t and (b) the angle Θ moved by a particle rotating with a constant angular velocity ω in time t.


Write an expression for (a) the constant linear acceleration a of a particle going from linear velocity vo at time t = 0 to a velocity v at time t and (b) the constant angular acceleration going from angular velocity ωo at time t = 0 to angular velocity ω  at time t.


Given that (a) dx/dt = v and a is constant, find x(t) for an object initially at xo traveling with vo at t = 0 and (b) dΘ/dt = w and αis constant, find Θ(t) for an object initially at Θo and rotating with ωo at t = 0.


Starting from Θo = 0 and ωo = π s-1, a wheel is given a constant angular acceleration of 4 π s-2.  After 4.0 s find (a) its angular speed and (b) how many turns the wheel has made.


The angular position of a particle is given by Θ(t) = b + ct + et2, where b, c, and e are constants. Find (a) the angular velocity, (b) the angular acceleration α, (c) the linear velocity, (d) the tangential acceleration, and (e) the centripetal acceleration of the particle at time t = t1. The particle travels in a circle of radius r.


The wheel shown in Fig. 1 below rotates with constant angular velocity ω. Show that points 1 and 2 on the wheel move with the same angular velocity, but different linear velocities. In Fig. 1, dark circles show the initial positions of the two points and the positions at time t are shown with open circles. Point 1 is a distance r1 from the axis and 2 a distance r2 > r1 from the axis.


Starting from rest at t = 0, a wheel undergoes a constant acceleration from t = 0 to t = 10 s. When t = 3.0 s, the angular velocity of the wheel is 6.0 rad/s. Through what angle does the wheel rotate from t = 0 to t = 20 s.


Some of the particles that make up the rigid body of a disk are shown in Fig. 2 . (a) Write a summation for the total kinetic energy of the disk with particles from i = 1 to i = N in terms of the mass mi of a particle and its velocity vi. (b) Substitute the expression for the angular velocity ω of the disk in terms of vi and ri. (c) Take the 1/2 and ω out of the summation and identify the quantity represented by the summation. (d) Rewrite the kinetic energy in terms of 1/2, ω and the quantity represented by the summation.


Two particles each of mass m are connected by two very thin rods of length L and rotate about axis O with a constant angular velocity ω as shown in Fig. 3 below. Find (a) the moment of inertia and (b) the kinetic energy of rotation of the system about O.


The object in Fig. 4 below is rotated about the axis at O by two forces. Find the magnitude and direction of the net torque on the object about O.
Given r1 = 2.0 m, F1 = 1.5 N, r2 = 1.0 m, F2 = 1.0 N.


A thin rod of length L and mass m is suspended at one end. It is pulled to one side and allowed to swing like a pendulum. It passes through the lowest point with an angular velocity ω. Find (a) its kinetic energy as it passes through its lowest position and (b) and the height to which its center of mass rises above its lowest position.


A pencil of length L, initially standing on one end falls over. With what speed does the eraser strike the horizontal surface, assuming the pencil point remains at rest on the surface? Consider the pencil to be a thin rod.


A uniform rod of length L and mass M is free to rotate about a frictionless pivot at the end attached to a wall. The rod is released from rest in the horizontal position. Find the initial (a) angular acceleration of the rod and (b) the linear acceleration of the right end of the rod.


Show that | τ | = rF sin r,F can be written as the product of the force times the component of r perpendicular to F.


You wish to push a wheel of mass M and radius R over a curb of height h by exerting a force F on the axle of the wheel. Find the minimum force F required to do the job.


A particle of mass m moves in the xy plane in a circular path of radius r. The motion of the particle is counterclockwise. Find (a) the magnitude and direction of its angular momentum relative to the center of the circle when its velocity is v and (b) an alternative expression for L in terms of the angular velocity ω.


Show that the magnitude of the angular momentum of a particle can be written as the product of (a) the momentum of the particle p and the component its position vector r perpendicular to p and (b) the position vector r and the component of its momentum p perpendicular to r.


Find the direction and magnitude of the angular momentum of the particle of mass m moving with velocity v a distance r from the axis shown in Fig. 6 below.


A light rod 1.0 m in length rotates in the xy plane about a pivot through the rod's center. Two particles of mass 4.0 kg and 3.0 kg are connected to its ends (Fig. 7 below). Determine the angular momentum of the system about the origin at the instant the speed of each is 5.0 m/s.


An artificial satellite is held in an elliptical orbit (Fig. 8) around the earth by the gravitational force Fg. Find the ratio of its velocity v1 at P1 a distance r1 from the earth to its velocity v2 at P2 a distance r2 from the earth. Explain your answer.


A sphere of radius R rolls down an inclined plane of height h and angle of inclination Θ. Find (a) using conservation of energy, the velocity of the center of mass when the sphere is at the bottom of the incline and (b) using Fnet = ma and τnet = Iα, (i) the acceleration of the center of mass, (ii) the frictional force that acts on the sphere, and (iii) the speed of the sphere at the bottom of the incline. Isphere about center of mass = 2/5 MR2.


A uniform ladder of length L and weight W = 50 N rests against a smooth, vertical wall. The smoothness of the wall means that the force due to the wall is horizontal and to the right, as shown in Fig. 9 below. If the coefficient of static friction between the ladder and ground is µs = 0.40, find the minimum angle Θmin such that the ladder will not slip.


A uniform board with a weight of 40 N and the applied forces shown in Fig. 10 below is in equilibrium. (a) Show all the other forces acting on the board and indicate their direction and magnitude. Explain your answer. (b) Take the axis at O, the position of the fulcrum, and show that the net torque acting on the board is zero. (c) Repeat (b) for an axis about O" at the left end of the board.


An object leans against a wall and remains at rest, as shown in Fig. 11 below. Its weight of 30 N acts at the center of gravity as shown in the figure. Find (a) the normal force FNormal, (b) the force of the wall on the object FWall on object
(c) the frictional force and (d) the coefficient of friction between the floor and the object.

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Susan D. Kunk
Phyllis J. Fleming
September 25, 2002
September 29, 2002