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Chapter 8: Q. 69 (page 203)

A small bead slides around a horizontal circle at height y inside the cone shown in FIGURE CP8.69. Find an expression for the bead’s speed in terms of a, h, y, and g.

Short Answer

Expert verified

The expression for velocity is v=ahgy

Step by step solution

01

Given Information

A small bead slides around a horizontal circle at height y inside the cone .

Radius of base of cone = a

Height of cone = h

02

Explanation

Lets draw Free Body diagram as below

From the diagram equation vertical and horizontal components of force

Rsin(θ)=mg..........................(1)and,Rcos(θ)=mv2r................................(2)

Divide equation(1) by (2) we get

mv2rmg=Rcos(θ)Rsin(θ)v2rg=tan(θ)..........................(3)

From the diagram we can get angle

ry=ahr=ayhtan(θ)=ah…………………………(4)

From equation (3) and (4)

v2=grtan(θ)v2=gayhahv2=a2gyh2v=ahgy

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