find speed

Question

The door of an automobile is open and perpendicular to the body. The automobile starts with an acceleration

of 2 ft/sec2

, and the width of the door is 30 inches. Treat the door as a uniform rectangle, and neglect

friction to find the speed of its outside edge as seen by the driver when the door closes.

Answer 1

This is actually a really fun problem, as if you try to think through the rotational and linear acceleration in the normal frame of reference, the math gets complicated in a hurry.  The last part should give us a hint that changing frame of reference will simplify things.  In fact, if we consider everything from the frame of reference of the driver, the problem rapidly simplifies.  From the driver's frame of reference, the door is experiencing an acceleration in the opposite direction of 2 ft/sec².  This then becomes a pendulum problem, although it is a rigid pendulum, not a simple pendulum.  So, I = mL²/3.  It starts with potential energy and no rotational kinetic energy.  It ends (instant before slamming shut) with zero potential and all rotational kinetic energy.  The potential energy lost is 1.25 ft * 2 ft/sec2 * m = 2.5m.  So, if we set the kinetic energy at the bottom to this potential energy, we get

 

Iω²/2 = 2.5m

mL²ω² / 6 = 2.5m

ω² = 2.5*6/(2.5)² = 2.4

ω = 1.55

 

Therefore, velocity of outside edge is angular velocity times radius or
1.55 * 2.5 = 3.875 ft/sec

 

If you have additional questions on that, please let me know.