SPRING 2002 QUIZ 4

PHYSICS 220A #85077

 

1. A 25.0-kg boy sits in a 55.0-kg-stationary boat. The boy throws a 5.3-kg package out horizontally with a speed of 9.0m/s. Find the velocity of the boat immediately after.

Solution.

From the conservation of momentum we have

                                (mboy+ mboat)vboat + mpackagevpackage = (mboy+ mboat)vboat' + mpackagevpackage'

                                 0 + 0 = (25.0kg + 55.0kg)vboat'  + (5.3kg)(9.0m/s),

which give vboat'  = -0.596m/s. (opposite to to the direction of the package).

 

2. A uniform thin-triangle plate of mass M is in the shape of a right triangle whose dimensions are shown in figure. Locate the x-coordinate of the center of mass,

 

 To evaluate the x coordinate of the center of mass, we divide the triangle into narrow strips of width dx and height y as in figure. We have

           dm = ρydx, where ρ = M/(1/2ab)= (2M/ab).

We find xCM from

           xCM = (òxdm)/M

                   = (1/M)ò0a (2M/ab)xydx

                   = (2/ab)ò0a xydx

From similar triangles in the figure, we see

             y/x = b/a, or y = (b/a)x.

We find 

               xCM = (2/ab)ò0a x(b/a)xdx

                       = (2/a2)ò0a x2dx

                       = (2/a2)[x3/3]|0a

                       = (2a)/3.