Smooth semisphere

Classical Mechanics Level 3

An object slides down a perfectly smooth fixed semisphere, starting from the top and with no initial speed. At what angle will it be no longer in contact with the semisphere?

Clarifications
The angle to find is the one formed between the object and the vertical axis (the angle when the object is at the top of the sphere is 0.)
Give the answer in degrees.
Give the answer to the second decimal place.


The answer is 48.18.

Sir isaac Newton by Sir isaac Newton , 65, Italy

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1 solution

Steven Chase
Nov 10, 2017

Gravity must supply the inward radial centripetal force. The point at which the object loses contact is the point at which the centripetal force equals the component of the gravitational force in the radial direction.

Conservation of energy:

m g ( R − R c o s θ ) = 1 2 m v 2 mg(R - R cos\theta) = \frac{1}{2} m v^2

Centripetal force:

m v 2 R = 2 m g ( 1 − c o s θ ) \frac{m v^2}{R} = 2 mg(1 - cos\theta)

Component of gravity in the radial direction:

F g r = m g cos ⁡ θ F_{gr} = mg \cos\theta

Equate the two:

m g cos ⁡ θ = 2 m g ( 1 − c o s θ ) 3 c o s θ = 2 θ = a c o s ( 2 3 ) ≈ 48.1 9 ∘ mg \cos\theta = 2 mg(1 - cos\theta) \\ 3 \, cos\theta = 2 \\ \theta = acos(\frac{2}{3}) \approx 48.19^\circ

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