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Converted document EXAMPLE: ROLLING CYLINDER WITH COUPLE

EXAMPLE: ROLLING CYLINDER WITH COUPLE

Consider a wheel undergoing the effects of opposite forces, as shown in the figure. The wheel has a mass of 10kg, a radius of 0.4m and a radius of gyration (normal to the plane for the figure), through C of 0.3m. The wheel is initially at rest and does not slip.
figure images/quiz6_figure.jpg
a) Draw and label the free body diagram.
b) Determine the angular acceleration of the wheel.
c) How far does the mass center, C, move in 3 seconds?
d) What is the velocity of C after it has moved 3m.
Solution
a) The free body diagram can be drawn as
figure images/quiz6_fbd.jpg
using a Cartesian coordinate system.
b) To determine the angular acceleration, it is first necessary to solve the equations of motion. There are two force and one angular equations of motion need to describe the two translations and one rotation. By definition the acceleration of the wheel is the angular velocity times the radius, c = .4α.
Fy  =  0 N  =  mg
Fx  =  mc 60 − 40 − f  =  10c 20 − f  =   − 4α
Mc  =  Icα 20(.4) − f(.4)  =  (.3)2α 8 − .4f  =  0.09α
Combining the force in the x direction and the moment equations yields:
8 − 0.4(20 + 4α)  =  0.09α 8 − 8 − 0.16α  =  0.09α 8  =  2.5α α  =  3.2(r)/(s2)
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