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How to Find the Center of Mass of a Cone

How to Find the Center of Mass of a Cone
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  • 0:04 Positioning the Cone
  • 1:12 Finding the Center of Mass
  • 5:30 The Ending Solution
  • 6:07 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we show how to find the center of mass of a cone. Of special interest is how to orient the cone in the Cartesian coordinate system to simplify the math.

Positioning the Cone

The center of mass of an object is the point about which the entire mass of the object is equally distributed. It's the point where the whole mass of the object may be considered to be concentrated.

If we look at a cone resting on a flat surface with its apex in the air, we can make reasonable statements about the location of the center of mass.


Cone resting on a flat surface
cone_resting_on_a_flat_surface


  • Symmetry: we expect the center of mass to be along a line from the center of the base to the apex. We will call this line the symmetry line.
  • Shape: There is more mass closer to the base. We expect the center of mass to be closer to the base than to the apex.

Without showing the details, if we worked this problem with the cone in this upright position, we would obtain the correct result but the integral would be more complicated than it has to be. How we position the cone will actually lead to simpler math. We will position the symmetry line along the x-axis and place the apex of the cone at the origin.


Symmetry line along x-axis
symmetry_line_along_x_axis


The equation for the center of mass, C.O.M., along the x-axis with respect to the origin is


COM_along_x-axis


Finding the Center of Mass

Step 1: Write the mass element in terms of a volume element.

The mass, M, is the product of the mass density, ρ, times the volume, V. Thus, the mass element, dM is


dM=rho_dV


Step 2: Substitute for dM in the C.O.M. equation.

Having an expression for dM we substitute


substituting_for_dM


For a uniform density, ρ does not depend on position and is a constant taken outside of the integral. The ρ in the numerator cancels with the ρ in the denominator:


afte_cancelling_the_rhos


Step 3: Replace the denominator with the volume of a cone.

The denominator is the integral of dV which equals the total volume. For a cone of height, h, and circular base whose radius is R, the total volume is


volume_of_a_cone


Note we are using this ''total volume of a cone'' result without deriving it.

Step 4: Substitute and simplify the C.O.M. equation.

Substitute the expression for the volume of a cone into the denominator of the C.O.M. equation.


substituting_for_V


Rearrange the expression so the 3 is in the numerator.


3_in_numerator


Step 5: Develop an expression for the volume element, dV

If we look along the z-axis towards the origin, we see a two-dimensional shape in the x-y plane.


2D_shape


The radius of the circle at the base is R. The height of the cone is h. At right-angles to the x-axis, we take a thin slice of thickness dx. Looking down the x-axis we see the slice as a circle with radius r.


Slice is a circle with radius r
circle_slice_with radius_r


The slice radius depends on x. When x = 0, the radius of the slice is 0. When x = h, the radius of the slice is R. Thus,


radius_of_the_slice


You can think of this as the equation of a line with slope R/h.

The volume of the slice is the area of the slice, πr2, times the width of the slice, dx. Thus,


dV=pir^2dx


Step 6: Simplify xdV

Now we work on simplifying the xdV, which we will need to integrate.

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