**Problem 72: Sliding Body on a Sphere**A body of mass \( m \) and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius \( R \). Prove that the body leaves the sphere when \( \theta = \cos^{-1}(2/3) \).**Diagram Explanation:**- The diagram shows a sphere with radius \( R \).- A small block is positioned at the top of the sphere and then depicted sliding down its surface.- The angle \( \theta \) is measured from the vertical centerline to the position of the block along the sphere's surface.- The point at which the block leaves the sphere is indicated by an arrow moving tangentially away from the sphere. This scenario is analyzed to find the angle \( \theta \) at which the normal force between the block and the sphere becomes zero. At this angle, the body loses contact and leaves the sphere’s surface.

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. A body of mass m and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius R. (See below.) Prove that the body leaves the sphere when 0 = cos¹ (2/3). R 0

. A body of mass m and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius R. (See below.) Prove that the body leaves the sphere when 0 = cos¹ (2/3). R 0

Physics for Scientists and Engineers: Foundations and Connections

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ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter7: Gravity

Section: Chapter Questions

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Concept explainers

Static and Kinetic Friction

Kinetic friction comes into existence when there are moving surfaces means in other words we can say that the force which comes into existing between moving surfaces is known as kinetic friction.

Question

**Problem 72: Sliding Body on a Sphere**

A body of mass \( m \) and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius \( R \). Prove that the body leaves the sphere when \( \theta = \cos^{-1}(2/3) \).

**Diagram Explanation:**

- The diagram shows a sphere with radius \( R \).
- A small block is positioned at the top of the sphere and then depicted sliding down its surface.
- The angle \( \theta \) is measured from the vertical centerline to the position of the block along the sphere's surface.
- The point at which the block leaves the sphere is indicated by an arrow moving tangentially away from the sphere.

This scenario is analyzed to find the angle \( \theta \) at which the normal force between the block and the sphere becomes zero. At this angle, the body loses contact and leaves the sphere’s surface.

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