A sled starts from rest at the top of a hemispherical?

Question

A sled starts from rest at the top of a hemispherical frictionless hill of radius R. (Actually
you can assume the sled has a very small velocity when at the top. This velocity is sufficient
to start the sled, but can be neglected for the rest of the calculations.)
(a) Use conservation of mechanical energy (plus a bit of trigonometry) to find an expression
for the sled’s speed when it is at angle .
(b) Use Newton’s 2nd law and ideas of circular motion to find the maximum speed the sled
can have at angle without leaving the surface. (Hint: the maximum speed is that at
which the normal force becomes zero.)
(c) Combine the results of (a) and (b) to find the numerical value of the angle max at which
the sled flies off the hill. This angle is independent of the mass of the sled, radius of the
hill, or the local value of g.

Answer 1

a)
Consider the energy of the sled:
It loses potential energy of m*g*R*(1-cos(a))
a is the angle that has opened from vertical

The kinetic energy is
1/2*m*v^2
so v=sqrt(2*g*R*(1-cos(a)))

for b
the normal force is
m*g*cos(a)
the centrifugal force is
m*v^2/R
v=sqrt(R*g*cos(a))

for c
using the maximum velocity for departure as
v^2=R*g*cos(a)
plug into
v^2=2*g*R*(1-cos(a))
R*g*cos(a)=2*g*R*(1-cos(a))
cos(a)=2(1-cos(a))
cos(a)=2/3
about 48 degrees


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