A solid small sphere of radius r and mass m starts from rest at a height H above

Question

A solid small sphere of radius r and mass m starts from rest at a height H above the low point of a track as shown above and roles down the track. No sliding occurs. After reaching the lowest point it continues along the track upward to a level plateau that is a height h above the lowest point of the track where it leaves horizontally from the vertical cliff. The vertical cliff is also at a height d above the ground below. The sphere hits the ground at a distance x from the base of the vertical cliff. What is mathematical equation that relates H ,h, x, and d?. The moment of inertia for a solid sphere is given in problem 6.

Explanation / Answer

let v is the speed of sphere at height h,

Apply Enrgy conservation

m*g*(H-h) = 0.5*m*v^2 + 0.5*I*w^2

m*g*(H-h) = 0.5*m*v^2 + 0.5*(2/5)*m*r^2*w^2

m*g*(H-h) = 0.5*m*v^2 + 0.2*m*v^2

m*g*(H-h) = 0.7*m*v^2

v = sqrt(g*(H-h)/0.7)

let t is the time taken to fall down

now Apply, d = voy*t + 0.5*g*t^2

= 0 + 0.5*g*t^2

==> t = sqrt(2*d/g)

x = v*t

= sqrt(g*(H-h)/0.7)*sqrt(2*d/g)

= sqrt(2*(H-h)*d/0.7)

= sqrt(20*(H-h)*d/7) <<<<---------Answer

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