A solid uniform disk of mass 21kg and radius 85cm is at rest flat on a frictionless surface. A string is wrapped around the rim of the disk and constant force of 35N is applied to the string, which does not slip on the rim, in the positive x direction. (a) In what direction does the center of mass move? When the disk has moved a distance of 5.5m ,determine (b )how fast it is moving, (c) how fast it is spinning,and (d) how much string has unwrapped from the rim.
To determine the direction of the center of mass' motion, we can use Newton's second law. The net force acting on the disk is the applied force on the string. According to Newton's second law, the direction of acceleration is determined by the net force acting on an object divided by its mass.
(a) In this case, the force applied to the string is in the positive x-direction. Therefore, the center of mass will also move in the positive x-direction.
To determine the speed at which the disk is moving after it has traveled a distance of 5.5m, we can use the work-energy theorem. The work done on an object is equal to the change in its kinetic energy.
(b) The work done on the disk is equal to the force applied multiplied by the distance moved (work = force x distance). The work is also equal to the change in kinetic energy of the disk. Using the equation for work:
Work = Force x Distance
35 N x 5.5 m = (1/2) x mass x velocity^2
Rearranging the equation, we can solve for the velocity:
Velocity = square root of ((2 x Work) / mass)
Substituting the known values:
Velocity = square root of ((2 x (35 N x 5.5 m)) / 21 kg)
By solving this equation, you will find the velocity at which the disk is moving after traveling a distance of 5.5m.
To determine how fast the disk is spinning, we can use the conservation of angular momentum. Angular momentum is the product of the moment of inertia and the angular velocity.
(c) When no external torque acts on an object, the angular momentum is conserved. Since there is no friction, the only torque acting on the disk is the torque produced by the applied force on the string. This torque causes the disk to rotate. The angular momentum is given by:
Angular momentum = moment of inertia x angular velocity
The moment of inertia for a solid disk rotating around its axis is (1/2) x mass x radius^2. Therefore, the angular velocity can be calculated as:
Angular velocity = (Angular momentum) / (moment of inertia)
Substituting the known values, you can calculate the angular velocity.
Finally, to determine the amount of string that has unwrapped from the rim, we need to find the arc length. The arc length can be calculated using the formula:
Arc length = (angle in radians) x radius
(d) Since the string is wrapped around the rim of the disk, the angle measured in radians is equivalent to the amount of string unwrapped from the rim. By using the formula for the arc length and substituting the known radius, you can calculate the amount of string that has unwrapped from the rim.
By following these steps, you should be able to find the answers to (a), (b), (c), and (d) of your question.