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
A solid cone of radius R, height H, and mass M.
The volume of a cone is 1/3 πHR2
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
A solid cone of radius R, height H, and mass M. The volume of a cone is 1/3 πHR2
Find the moment of inertia using dm from equation (2)
So moment of inertia is
Now integrate to find moment of inertia
Your task in a science contest is to stack four identical uniform bricks, each of length L, so that the top brick is as far to the right as possible without the stack falling over. Is it possible, as Figure P12.61 shows, to stack the bricks such that no part of the top brick is over the table? Answer this question by determining the maximum possible value of d.
A cylinder of radius R, length L, and mass M is released from rest on a slope inclined at angle θ. It is oriented to roll straight down the slope. If the slope were frictionless, the cylinder would slide down the slope without rotating. What minimum coefficient of static
friction is needed for the cylinder to roll down without slipping?
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