=========================preview======================
(PHYS221)[2010](f)quiz~2678^_10048.pdf
Back to PHYS221 Login to download
======================================================
PHYS221
Intermediate Classical Mechanics

Solution of Quiz 1




(a) First you need to note that the track can only push on the block and cannot pull the block towards the track.

During the passage from P to Q, the component of the gravity points outwards and away from the center of curvature. Thus in this section, the track pushes on the block to keep it moving along the track and the block is in contact with the track.

When the block moves past Q and towards the top of the loop, the component of gravity now points towards the center. The magnitude of this component is zero at Q and increases as where is the angle that the block makes with the vertical. When the block gets to Q, its speed is obviously non-zero (from energy conservation) and it is thus undergoing centripetal acceleration towards the center. At this point, the track provides the needed force in the centripetal direction by pushing on the block. Thus the block is still in contact with the track at Q. As the block continues to move upward, its speed decreases while the component of the gravity towards the center increases. At a certain angle, the speed of the block will be such that the component of gravity in the centripetal direction is equal to the mass of the block times the centripetal acceleration. At this angle, the force on the block due to the track is zero. When the block continues to move higher, its speed slows further while the component of gravity increases further and the block will be pulled away from the track.




The above physical reasoning can be summarized in the following equations.