ME 230 - Dynamics



ME 230 - Dynamics Your Name:_________________

Tutorial 6 Section No.:_________________

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Angular Momentum and 2-D Kinematics

Relationships of Interest

Angular Momentum: [pic] and [pic]

Rate of Change of Angular Momentum [pic]

Conservation of Angular Momentum: [pic] or [pic]

1) The mechanical governor shown consists of two spheres of identical masses m mounted on two rigid rods of negligible mass and of length L, rotating freely about a vertical shaft. An internal mechanism adjusts the angular position ( of the rods to secure the desired angular velocity of the shaft. If the angular velocity of the spheres is given as (o when the rods are horizontal ((o = 90o), determine its magnitude for any angle ( in terms of (o , then find its magnitude when ( = 45o.

Draw a coordinate reference frame on the diagram above, in cylindrical coordinates, with the origin at O.

Is angular momentum conserved about the z axis? Why? Write the symbolic statement of conservation of angular momentum about the z axis.

Write an expression for the angular momentum Ho at (o = 90o in terms of m, L and (o.

Write a similar expression the angular momentum Ho at any other angle ( in terms of m, L, (, and (.

Equate the expressions for angular momentum at 90o and at any angle (. Why can you do this? Obtain a general expression for the angular velocity at any position ( in terms of the angular velocity at (o (when ( = 90o).

From the above expression, what is the angular velocity of the shaft at ( =45o.

2) The cylinder of diameter R rolls without slipping on the plane surface. Point A is moving to the right a constant speed vA. What is the angular velocity vector of the cylinder. Find the velocity and acceleration of points B, C D, and E. (similar to problems 6.27 and 6.85)

What is the angular velocity vector for the disk?

Write the position vector symbolically for points B, C, D, and E in terms of rA = R j, where R is the cylinder radius, in terms of R and (.

Take the time derivative of each position vector to find the velocity vector of each point.

Roll the cylinder across the table, noting the path described by a point on its rim. In the space below, trace out the path described by that point and label the points corresponding to B, C, D, and E. Draw small arrows representing the velocity of each of the four points.

Take the time derivative of each velocity vector to find the acceleration vector of each point.

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