Resource Lesson
Thin Rods: Center of Mass
Printer Friendly Version
Let's derive the equation needed to calculate the center of mass of a uniform, continuous rod. Remember that the equations used to calculate the center of mass of a collection of discreet masses were
and
The center of mass in the vertical dimension, y
_{cm}
, would be somewhere along the rod's central axis and will be included at the end of the derivation. For now, we will focus on calculating the position of the rod's horizontal center of mass, x
_{cm}
.
Since the rod is a continuous mass, we must add up all of the small mass segments,
Δm
_{i}
, located at a distance
x
_{i}
from the left hand side to calculate its center of mass.
When we take the limit of this summation as
Δm
approaches zero we obtain a general expression for the center of mass.
But we can not integrate
x
with respect to
dm
, so we must first develop a
relationship between dm and x.
This is done with a quantity called
λ, or the mass per unit length
. For a uniform rod, λ would equal a constant value. For example, a rod might have a mass per unit length of λ = 0.4 kg/m.
Substituting
λdx
for
dm
now allows us to integrate to calculate the rod's center of mass.
Note that the expression
λL
equals the mass of the rod,
M
; that is, when we multiple the mass per unit length by the length of the rod we obtain the rod's entire mass.
This result, that the center of mass which was painstakingly obtained would have probably been your first logical guess. The center of mass of a uniform rod is at the center of the rod. But what if the mass per unit length, λ, was not uniform, but also a function of x? Then we would need the process outlined above to obtain our answer. Let's look at two more complicated examples.
Where would the center of mass be located for a continuous body that has a mass per unit length distribution of λ = Ax (this rod would look like a rough "baseball bat")?
Where would the center of mass be located for a continuous body that has a mass per unit length distribution of λ = Ax
^{2}
(this rod would look like a rough "trumpet bell")?
Related Documents
Lab:
Labs -
A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus
Labs -
Conservation of Momentum in Two-Dimensions
Labs -
Density of an Unknown Fluid
Labs -
Mass of a Paper Clip
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Rotational Inertia
Resource Lesson:
RL -
A Chart of Common Moments of Inertia
RL -
A Further Look at Angular Momentum
RL -
Average Velocity - A Calculus Approach
RL -
Center of Mass
RL -
Centripetal Acceleration and Angular Motion
RL -
Derivatives: Instantaneous vs Average Velocities
RL -
Discrete Masses: Center of Mass and Moment of Inertia
RL -
Hinged Board
RL -
Introduction to Angular Momentum
RL -
Rolling and Slipping
RL -
Rotary Motion
RL -
Rotational Dynamics: Pivoting Rods
RL -
Rotational Dynamics: Pulleys
RL -
Rotational Dynamics: Rolling Spheres/Cylinders
RL -
Rotational Equilibrium
RL -
Rotational Kinematics
RL -
Rotational Kinetic Energy
RL -
Thin Rods: Moment of Inertia
RL -
Torque: An Introduction
Worksheet:
APP -
The Baton Twirler
APP -
The See-Saw Scene
CP -
Center of Gravity
CP -
Torque Beams
CP -
Torque: Cams and Spools
NT -
Center of Gravity
NT -
Center of Gravity vs Torque
NT -
Falling Sticks
NT -
Rolling Cans
NT -
Rolling Spool
WS -
Moment Arms
WS -
Moments of Inertia and Angular Momentum
WS -
Practice: Uniform Circular Motion
WS -
Rotational Kinetic Energy
WS -
Torque: Rotational Equilibrium Problems
TB -
Antiderivatives and Kinematics Functions
TB -
Basic Torque Problems
TB -
Center of Mass (Discrete Collections)
TB -
Moment of Inertia (Discrete Collections)
TB -
Projectile Summary
TB -
Projectile Summary
TB -
Rotational Kinematics
TB -
Rotational Kinematics #2
PhysicsLAB
Copyright © 1997-2017
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton