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9 problems tagged with rotational motion
P0340
Advanced Mechanics › KinematicsAcceleration of Contact Point on Rolling Wheel
As shown in Figure, a wheel with radius $r$ rolls without slipping on the outer surface of a fixed cylinder with radius $R$. The center of the wheel, O, moves with a constant speed $V$.
P0320
Beginner Mechanics › Rotational MotionFlywheel Angular Deceleration and Motion Analysis
The angular velocity of a flywheel uniformly decreases from 900 rev/min to 800 rev/min in 5 seconds.
- Find the angular acceleration β.
- Find the total number of revolutions during these 5 seconds.
- Find how many more seconds it will take for the wheel to stop rotating.
P0331
Advanced Mechanics › KinematicsKinematics of a Point on a Purely Rolling Ring
A rigid ring with radius $R$ undergoes pure rolling on a rigid horizontal surface. The center of the ring moves forward horizontally with a constant velocity $v_0$. Consider a point P on the ring that is at the same height as the center.
- Find the instantaneous velocity of point P.
- Find the tangential acceleration of point P.
- Find the normal acceleration of point P.
P0622
Beginner Mechanics › Rotational MotionAngular Velocities of Earth's Motion
Calculate the angular velocities of the Earth's daily rotation about its axis and its annual revolution about the Sun.
- What is the angular velocity of the Earth's rotation?
- What is the angular velocity of the Earth's revolution?
P0623
Beginner Mechanics › Rotational MotionUniformly Accelerated Rigid Body Rotation
A rigid body undergoes uniformly accelerated rotation about a fixed axis. The angular acceleration is $\alpha = 2.0 \text{ rad/s}^2$, and the initial angular velocity at $t=0$ is $\omega_0 = 4 \text{ rad/s}$.
- What is the angular velocity of the rigid body at $t = 3 \text{ s}$?
- How many revolutions does the rigid body turn through in the first $3 \text{ s}$?
- At $t = 3 \text{ s}$, what are the linear velocity and centripetal acceleration of a point $10 \text{ cm}$ from the axis of rotation?
P0624
Beginner Mechanics › Rotational MotionFlywheel Uniform Angular Deceleration
A flywheel with a rotational speed of 1500 r/s decelerates uniformly after braking and comes to a stop in 5.0 s. The diameter of the flywheel is 1.00 m.
- What is the angular acceleration of the flywheel?
- How many revolutions does the flywheel make from the start of braking until it stops?
- What is the angular velocity of the flywheel 2.5 s after braking begins?
- What is the linear velocity of a point on the edge of the flywheel 2.5 s after braking begins?
P0625
Beginner Mechanics › Rotational MotionMoment of Inertia of a Dumbbell
A thin rod of length 80 cm has a 200 g small ball attached to each end. The balls can be treated as point masses, and the mass of the rod is negligible.
- What is the moment of inertia of the setup when it rotates about an axis passing through the center of the rod and perpendicular to it?
- What is the moment of inertia of the setup when it rotates about an axis passing through one of the balls and perpendicular to the rod?
P0718
Expert Mechanics › KinematicsUnwinding Spool
A spool consists of two outer disks of radius $R$ and an inner hub of radius $r$. The total mass is $M$, and the moment of inertia about the center of mass is $I$. A massless string is wrapped around the inner hub in a counter-clockwise fashion. You pull the string with a tension force $T$ at an angle $\theta$ with the horizontal (the string is pulled from the bottom side of the inner hub). The spool rolls without slipping on a rough horizontal floor.
- Depending on the angle $\theta$, the spool can roll to the right (unwinding) or to the left (winding up). What is the geometric condition that determines the direction. Calculate the exact angle $\theta_{crit}$ at which the spool does not roll (i.e., the transition point between forward and backward motion).
- If the string is pulled horizontally ($\theta = 0^\circ$), find the linear acceleration $a$ of the center of mass and the angular acceleration. Assume rolling without slipping.
- If the string is pulled at angle $\theta$ with horizontal, find the linear acceleration $a$ of the center of mass and and the angular acceleration. Assume rolling without slipping.
- You pull the string with constant force $T$ (horizontal, from underside). The center of mass moves a distance $L$. How much work did you do? and what is the displcement and rotational kinetic energy of the system?
- When the string is pulling at constant speed $v$ at an angle $\theta$ with respect to horizontal, the spool is moving forward. Calculate the speed at which the spool is moving.
P0777
Advanced Mechanics › Waves and SoundPlank Oscillating on Counter-Rotating Rollers
Two cylinders of different radii, $R$ and $r$, rotate in opposite directions about parallel horizontal axes with the same angular velocity $\omega$. The axes are separated by a distance $L$. At $t=0$, a uniform plank of mass $M$ is placed horizontally on the rollers, with its center of mass (CM) initially above the axis of one of the cylinders. The coefficient of kinetic friction is $\mu$. The top surfaces of the rollers move inwards.
- Find the equation of motion for the horizontal displacement $x$ of the plank's CM from the midpoint between the rollers.
- Find the plank's horizontal displacement as a function of time, $x(t)$.
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