IIn this problem, consider the figure below. A disk with radius r, mass m, and moment of inertia of Io about the point O ison an inclined plane, with the translational coordinates and y and rotation angle 0, as shown. The direction of gravity isshown, and the incline angle is a. A constant input torque is applied at the point O in the direction shown, and it is opposedby a damping torque from a bearing with value co in the direction shown. The point P represents a point of instantaneouscontact, rolling without slipping.Your tasks:Direction ofGravityM(t)co0PxMass: mFigure 2: System schematic.Moment of Inertiaabout 0: loa: Incline AngleA Draw the FBD for the disk.B Derive the equation of motion with the rotation angle as the dynamic variable.0C Take the Laplace transform, and solve for (s), letting the initial conditions be 0(0) = 0 and (0) = 0. Remember thata constant is a step function for a one-sided Laplace transform.D Given your expression for (s), find the Laplace transform of the angular velocity, (s)E Assuming the plane is quite long, what is the terminal angular velocity (the final value of the angular velocity) of thedisk, was?

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In this problem, consider the figure below. A disk with radius r, mass m, and moment of inertia of Io about the point O is on an inclined plane, with the translational coordinates and y and rotation angle 0, as shown. The direction of gravity is shown, and the incline angle is a. A constant input torque is applied at the point O in the direction shown, and it is opposed by a damping torque from a bearing with value ce in the direction shown. The point P represents a point of instantaneous contact, rolling without slipping.

In this problem, consider the figure below. A disk with radius r, mass m, and moment of inertia of Io about the point O is on an inclined plane, with the translational coordinates and y and rotation angle 0, as shown. The direction of gravity is shown, and the incline angle is a. A constant input torque is applied at the point O in the direction shown, and it is opposed by a damping torque from a bearing with value ce in the direction shown. The point P represents a point of instantaneous contact, rolling without slipping.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Generally, there are three types of vibration: longitudinal vibration, transverse vibration, and torsional vibration.

Question

I
In this problem, consider the figure below. A disk with radius r, mass m, and moment of inertia of Io about the point O is
on an inclined plane, with the translational coordinates and y and rotation angle 0, as shown. The direction of gravity is
shown, and the incline angle is a. A constant input torque is applied at the point O in the direction shown, and it is opposed
by a damping torque from a bearing with value co in the direction shown. The point P represents a point of instantaneous
contact, rolling without slipping.
Your tasks:
Direction of
Gravity
M(t)
co
0
P
x
Mass: m
Figure 2: System schematic.
Moment of Inertia
about 0: lo
a: Incline Angle
A Draw the FBD for the disk.
B Derive the equation of motion with the rotation angle as the dynamic variable.
0
C Take the Laplace transform, and solve for (s), letting the initial conditions be 0(0) = 0 and (0) = 0. Remember that
a constant is a step function for a one-sided Laplace transform.
D Given your expression for (s), find the Laplace transform of the angular velocity, (s)
E Assuming the plane is quite long, what is the terminal angular velocity (the final value of the angular velocity) of the
disk, was?

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