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 3
 Homework Statement

A perfect hemisphere of frictionless ice has radius R=6.5 meters. Sitting on the top of the ice, motionless, is a box of mass m=6 kg.
The box starts to slide to the right, down the sloping surface of the ice. After it has moved by an angle 20 degrees from the top, how much work has gravity done on the box?
How fast is the box moving?
 Homework Equations
 g=9.8m/s^2
Homework Statement: A perfect hemisphere of frictionless ice has radius R=6.5 meters. Sitting on the top of the ice, motionless, is a box of mass m=6 kg.
The box starts to slide to the right, down the sloping surface of the ice. After it has moved by an angle 20 degrees from the top, how much work has gravity done on the box?
How fast is the box moving?
Homework Equations: g=9.8m/s^2
From the force diagram, Fn=mgcos(theta) and m(ax)=mgsin(theta). Net force=m(ax). Thus the integral of net force=mgsin(theta)*(distance traveled). Since the circumference of half a sphere = 2(pie)(6.5m)/2 = 20.42m, the distance traveled= (20.42m)*(20degree)/(180degree)= 2.27m. The final equation should be: work=(6kg)(9.8m/s^2)sin(20deg)*(2.27m)=45.65J.
The box starts to slide to the right, down the sloping surface of the ice. After it has moved by an angle 20 degrees from the top, how much work has gravity done on the box?
How fast is the box moving?
Homework Equations: g=9.8m/s^2
Force  x  y 
normal force of sphere on box  0  Fn 
gravity  mgsin(theta)  mgcos(theta) 
total  m(ax)  m(ay)=0 
From the force diagram, Fn=mgcos(theta) and m(ax)=mgsin(theta). Net force=m(ax). Thus the integral of net force=mgsin(theta)*(distance traveled). Since the circumference of half a sphere = 2(pie)(6.5m)/2 = 20.42m, the distance traveled= (20.42m)*(20degree)/(180degree)= 2.27m. The final equation should be: work=(6kg)(9.8m/s^2)sin(20deg)*(2.27m)=45.65J.