Mud from Tire
المؤلف:
Sidney B. Cahn And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 7
28-7-2016
1862
Mud from Tire
A car is stuck in the mud. In his efforts to move the car, the driver splashes mud from the rim of a tire of radius R spinning at a speed v, where v > gR Neglecting the resistance of the air, show that no mud can rise higher than a height R+ v2 /2g+ gR2/2v2 above the ground.
SOLUTION
Mud flying from different points on the tire will rise to different heights, depending on the initial height and angle of ejection. Introducing an angle α and the height h of the point of ejection O above the equator of the tire (see Figure 1.1), we can write using energy conservation
where h0 is the height to which the mud rises above O, and v = ωR is the speed of the rim of the wheel. The height H above the ground will be
Figure 1.1
Now find the maximum height by setting the derivative of H with respect to α equal to 0:
There are two solutions of this equation. First,
(1)
This case yields a maximum only when v2 ≤ gR, and the highest point of the wheel is the maximum height. But here v2 > gR, so we will consider the other case:
(2)
The height becomes
We can check that
الاكثر قراءة في مواضيع اخرى
اخر الاخبار
اخبار العتبة العباسية المقدسة
