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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.

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

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, ẍ_c = .4α.

⎲⎳F_y = 0 N = mg

⎲⎳F_x = mẍ_c 60 − 40 − f = 10ẍ_c 20 − f = − 4α

⎲⎳M_c = I_cα 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)