A small mass m starts from rest and slides down the surface of a frictionless solid sphere of radius R. Calculate the angle at which the mass flies off the sphere. If there is friction between the mass and the sphere, does the mass of fly off at a greater or lesser angle than the previous one ?

Explanation

To solve the problem step by step, we will analyze the motion of the mass \( m \) sliding down the frictionless solid sphere and find the angle at which it flies off.

### Step 1: Understanding the Forces Acting on the Mass
When the mass \( m \) is at an angle \( \theta \) from the vertical, two forces act on it:
1. The gravitational force \( mg \) acting downwards.
2. The normal force \( N \) acting perpendicular to the surface of the sphere.

### Step 2: Setting Up the Equations of Motion
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