Chapter 5: Q18MP (page 274)

Find the center of mass of the solid right circular cone inside $r^{2} = z^{2}, 0 < z < h$ , If the density is $r^{2} = x^{2} + y^{2}$ . Use cylindrical coordinates.

Short Answer

Expert verified

The center of mass of the solid right circular cone inside $r^{2} = z^{2}, 0 < z < h$ with density $r^{2} = x^{2} + y^{2}$ in cylindrical coordinates is $(x, y, z) = (0, 0 \frac{5}{6} h) .$

Step by step solution

Given Condition

The solid right circular cone is $r^{2} = z^{2}, 0 < z < h$ . Its density is $r^{2} = x^{2} + y^{2}$ .

Concept of center of mass

The center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration.

Draw the diagram

Calculate the center of mass

The mass is given as follows

$M = \int_{0}^{2 π} d θ \int_{0}^{h} d z \int_{0}^{z} r d r r^{2} = \frac{π}{2} \int_{0}^{h} d z z^{4} = \frac{π}{10} h^{5}$

We see that $x = y = 0$

So, $z$ is

$M z = \int_{v} ρ z d V = \int_{0}^{2 π} d θ \int_{0}^{h} z d z \int_{0}^{z} r^{3} d r = \frac{π}{2} \int_{0}^{h} z d z z^{4} = \frac{π}{12} h^{6}$

Calculate the coordinates

Thus, $z = \frac{5}{6} h$

Therefore, the coordinates are $(x, y, z) = (0, 0 \frac{5}{6} h)$

Hence, the center of mass of the solid right circular cone inside $r^{2} = z^{2}, 0 < z < h$ with density $r^{2} = x^{2} + y^{2}$ in cylindrical coordinates is $(x, y, z) = (0, 0 \frac{5}{6} h)$ .

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Find the center of mass of the solid right circular cone inside $r^{2} = z^{2}, 0 < z < h$ , If the density is $r^{2} = x^{2} + y^{2}$ . Use cylindrical coordinates.

Short Answer

Step by step solution

Given Condition

Concept of center of mass

Draw the diagram

Calculate the center of mass

Calculate the coordinates

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Most popular questions from this chapter

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Find the center of mass of the solid right circular cone inside r2=z2,0<z<h, If the density isr2=x2+y2. Use cylindrical coordinates.

Short Answer

Step by step solution

Given Condition

Concept of center of mass

Draw the diagram

Calculate the center of mass

Calculate the coordinates

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Electricity and Magnetism

Solid State Physics

Translational Dynamics

Electricity

Fluids

Study anywhere. Anytime. Across all devices.

Find the center of mass of the solid right circular cone inside $r^{2} = z^{2}, 0 < z < h$ , If the density is $r^{2} = x^{2} + y^{2}$ . Use cylindrical coordinates.