Respuesta :
Answer:
48.189°
Explanation:
Let us say,
Radius of curvature of the clown's bald head is = R
Angle where the grape leaves the contact with the head is (with vertical) = θ
Height from the top of the head at which the contact is lost = y
Mass of the grape = m
Velocity of the grape at the point where it loses contact = v
So,
Using the Conservation of Work and Energy, we can say that there is 0 Work done on the system,
W = ΔK + ΔU
So,
[tex]0=(\frac{1}{2}mv^{2}-0)+(mgy-0)\\v^{2}=2gy\\Now,\\y=R-Rcos\theta\\y=R(1-cos\theta)\\So,\\v^{2}=2[R(1-cos\theta)]g[/tex]
Now, using this at the point where contact is lost,
[tex]N=-m(\frac{v^{2}}{R})+mg.cos\theta\\N=-m[2g(1-cos\theta)]+mg.cos\theta\\[/tex]
At that point the Normal force will be zero, because the contact is lost.
So,
On putting, N = 0 we get,
[tex]N=-m[2g(1-cos\theta)]+mg.cos\theta\\0=-m[2g(1-cos\theta)]+mg.cos\theta\\2g-2g.cos\theta=g.cos\theta\\3g.cos\theta=2g\\cos\theta=\frac{2}{3}\\\theta=48.189\,degrees[/tex]
Therefore, the angle at which the grape lose contact with the bald head is at 48.189° from vertical.
