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Q. 56

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

Consider a solid cone of radius R, height H, and mass M. The volume of a cone is 1/3 πHR2

a. What is the distance from the apex (the point) to the center of mass? b. What is the moment of inertia for rotation about the axis of the cone? Hint: The moment of inertia can be calculated as the sum of the moments of inertia of lots of small pieces.

a) Center of mass is at 3/4 H from vertex.

b) The moment of inertia is 3Mr2/10

See the step by step solution

Step by Step Solution

Part(a) Step 1: Given

A solid cone of radius R, height H, and mass M.

The volume of a cone is 1/3 πHR2

Part(a) Step2: Explanation

Lets consider a differential disc with radius r at a distance of x from the vertex as shown in figure below.

The mass of the differential disc is dm.

density of the disk is can be calculated as

From the figure we can find

Now we can find the value of dm

Substitute the value we get

Now find the center of mass

Part(b) Step 1: Given information

A solid cone of radius R, height H, and mass M. The volume of a cone is 1/3 πHR2

Part(b) Step 2: Explanation

Find the moment of inertia using dm from equation (2)

So moment of inertia is

Now integrate to find moment of inertia

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