ME 230 - Dynamics



ME 230 - Dynamics Your Name:_________________

Tutorial 9 Section No.:_________________

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Energy and Momentum Methods for Rigid Body Motion

Relationships of Interest:

For general planar motion: [pic]

Work and Energy: [pic]

Conservation of Energy: [pic]

I for Slender Rod: About C.M., [pic] About end, [pic]

Potential Energy: For gravity, [pic], For spring, [pic] (S = stretch, [pic])

Linear Impulse and Momentum: [pic]

Angular Impulse and Momentum: [pic], where [pic]

1) A 4-ft. slender bar weighs 40 lb. with mass center at B and is released from rest in the position for which [pic] is essentially zero. Point B is confined to move in the smooth vertical guide, while end A moves in the smooth horizontal guide and compresses the spring as the bar falls. Determine a) the angular velocity of the bar as position [pic] = 30o is passed and b) the velocity with which B strikes the horizontal surface if the stiffness of the spring is 30 lb./in.

Part a)

i. Are the forces conservative? If so, write the symbolic statement of conservation of energy. Then write the symbolic statement of kinetic energy for a rigid body. Is the symbolic statement the same if point A is used as the reference point rather than the center of mass?

ii. For the first 30 degrees of motion, is the spring involved?

iii. Determine the kinematic relation between the velocity [pic] of the center of mass and the angular velocity [pic] of the bar. Do this by finding the instantaneous center and recalling that [pic].

iv. Find the kinetic energy of the system at [pic] = 30o ,and using conservation of energy, solve for the angular velocity of the bar at that point.

Part b)

v. Find the kinetic energy for the rod when it reaches the horizontal position.

vi. Now consider the motion of the rod. Find the potential energy for the rod at the initial position and the final position when the rod strikes the horizontal surface.

vii. Using conservation of energy, find the velocity with which B strikes the horizontal surface

2) A 30 g bullet is fired with a horizontal velocity of 400 m/s into a 4 kg wooden beam AB suspended from a pin support at A. Knowing that h = 600 mm and that the beam is initially at rest, determine a) the velocity of the mass center G of the beam immediately after the bullet becomes embedded, and b) the reactions at A, assuming that the bullet becomes embedded in 1 ms.

i. Define your system to be the rod and the bullet. Draw a free body/momentum diagram of the system to illustrate the linear and angular impulse-momentum equations for an impulse equal to the change in momentum. The diagram should include a figure representing the period during which the bullet becomes embedded (showing the external forces on the rod), and figures showing the system just after and just before the bullet becomes embedded.

ii. Write expressions for angular impulse and momentum and linear impulse and momentum pertaining to the three cases above. You may neglect the mass of the bullet.

iii. Determine the kinematic relationship between the velocity of the center of mass, [pic], and the angular velocity of the bar, [pic].

iv. From these three relationships, solve for [pic], [pic], and the reaction(s) at A. Does the reaction at A in the vertical direction change because of the impact?

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