Circular motion of flywheel axis (precession) Flywheel Rotation of flywheel Pivot y
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Q: What is the angular momentum of the disk after those rotations are complete, in kg m2/s?
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A gyroscope consists of a flywheel of mass m, which has a moment of inertia I for rotation about its axis. It is mounted on a rod of negligible mass, which is supported at one end by a frictionless pivot attached to a vertical post, as shown in the diagram. The distance between the center of the wheel and the pivot is d. The wheel rotates about its axis with
Adopt a coordinate system with the z axis pointing upward and the x and y axes in the horizontal plane. The gyroscope is moving, but at t=0, the rod is in the yz plane.
The gyroscope is observed to precess about the vertical axis, with an angular velocity of precession Ω, defined as positive for counterclockwise precession as seen from above. Find Ω in terms of the given quantities.
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- Problem 2: A pulley consists of a large disk of radius R and a small disk of radius r that are welded together and mounted on a horizontal frictionless axle through their common centers. The moment of inertia of this pulley is I. A box containing a turkey of mass M is suspended from a rope wound around the large disk; a box containing a pumpkin of mass m is suspended from a rope wound around the small disk. The ropes do not slip on the disks. The system is released from rest and the turkey begins to descend, while the pumpkin is being lifted up. M P A Derive an expression for the magnitude of the angular acceleration a of the pulley, in terms of system parameters. Assume that a counterclockwise motion of the pulley is the positive direction. MgR+mgR MgR-mgr MgR-mgR a a = a = I+MR2+mr2 I+MR²+mr² I+MR²+mR? MgR+mgr MgR-mgR MgR+mgR a = a = I+MR2+mr²2 I+MR²+mr² I+MR²+mR? Submit I give up! Hint FeedbackA turntable (disk) of radius r = 27.0 cm and rotational inertia 0.440 kg · m2 rotates with an angular speed of 2.98 rad/s around a frictionless, vertical axle. A wad of clay of mass m = 0.242 kg drops onto and sticks to the edge of the turntable. What is the new angular speed of the turntable? (A) _______ rad/sThe figure above shows two barbells, which can be modeled as two point masses connected by a massless rod. In the figure, L = 0.77 m and m = 20.1 kg. The barbells can rotate about an axis that is perpendicular to the rod and passes through the midpoint of each rod. For the barbell with the larger moment of inertia, give the moment of inertia (in units of kg m2).
- A solid cylinder with mass m, radius R, and rotational inertia I (about its center) is released from rest and rolls down a ramp. Friction between the bottom of the cylinder and the ramp causes the cylinder to roll without slipping. The linear acceleration of the cylinder is a. Which TWO of the equations below are correct for this scenario? The equations are in terms of m, I, R, and a, as well as f (the force of static friction between the cylinder and the ramp) and g. (You must pick BOTH of them to get this question correct!) f = ma mg sin θ = ma mg sin θ - f = ma f R = I (a/R) mg sin θ - f = I (a/R) R mg sin θ = IaRIn the figure here, a 29 kg child stands on the edge of a stationary merry-go-round of radius 2.2 m. The rotational inertia of the merry- go-round about its rotation axis is 100 kg⋅m². The child catches a ball of mass 0.9 kg thrown by a friend. Just before the ball is caught, it has a horizontal velocity of magnitude 9 m/s, at angle = 38° with a line tangent to the outer edge of the merry-go-round, as shown. What is the angular speed of the merry-go-round just after the ball is caught? W = i Child BallTwo disks are initially spinning, one above the other on a small axle that provides a small, but non-negligible torque from friction, as shown in the figure below. Both disks have the same radius, R = 2.58 m. Disk 1 has a moment of inertia I1 = 9.8 kg⋅m2. Disk 2 has a moment of inertia I2 = 5 kg⋅ m2. Let vertically up be the z direction, such that counterclockwise rotation as viewed from above corresponds to positive values of the z-component. Disk 1 is initially spinning with a z-component of angular velocity ω1,z = 21 rad/s, and disk 2 is initially spinning with a z-component of angular velocity ω2,z = -15 rad/s. The z component of their common angular velcoity is 8.837 rad/s How much thermal energy is created in the process of disk 1 falling on disk 2 such that they reach a common final angular velocity? You do not need to worry about the gravitational potential energy because the initial separation of the disks is small. I get I need to use kinetic energy equations of 1/2*I*w2 but…
- A uniform horizontal disk of radius 5.50 m turns without friction at w = 2.30 rev/s on a vertical axis through its center, as in the figure below. A feedback mechanism senses the angular speed of the disk, and a drive motor at A ensures that the angular speed remain constant while a m = 1.20 kg block on top of the disk slides outward in a radial slot. The block starts at the center of the disk at time t = 0 and moves outward with constant speed v = 1.25 cm/s relative to the disk until it reaches the edge at t = 465 s. The sliding block experiences no friction. Its motion is constrained to have constant radial speed by a brake at B, producing tension in a light string tied to the block. (a) Find the torque as a function of time that the drive motor must provide while the block is sliding. Hint: The torque is given by = 2mrvw. t N-m (b) Find the value of this torque at t= 465 s, just before the sliding block finishes its motion. N.m 2.52 (c) Find the power which the drive motor must…The figure shows three 0.0114 kg particles that have been glued to a rod of length L= 5.70 cm and negligible mass. The assembly can rotate around a perpendicular axis through point O at the left end. If we remove one particle (that is, 33% of the mass), by what percentage does the rotational inertia of the assembly around the rotation axis decrease when that removed particle is (a) the innermost one and (b) the outermost one? Axis m m (a) Number i Units (b) Number i Units >A meter stick is attached at 1 end to a pivot and the other end is allowed to drop. The m stick starts out parallel to the floor but will be allowed to fall so that it will be vertical. How long will it take for the free end of the meter stick to fall to vertical position? What is the angular acceleration of the meter stick? (hint – The moment of inertia of the stick that is I= 1/12 m l^2, mass is 450 g and clearly length is 1 m.) )rotating about 1 end. Use a conservation of energy statement to find the final angular velocity of the stick. Use this to determine it’s angular acceleration.) What is the tangential or linear speed of the free end of the meter stick once it has reached vertical?