The translational velocity
If the object is rolling without slipping, the translational velocity of the center of mass is given by
Vcm = W*R
where
- Vcm represents the translational velocity of the center of mass
- W represents the angular velocity
- R represents the radius
A well-defined relationship
This means that there is a well-defined relationship between the rolling object's translational and rotational kinetic energies. The totalkinetic energy of a rolling object is the sum of its translational and rotational kinetic energies.
A clue
This may give you a clue as to where we are heading with this. As a preview, when an object rolls down an incline it exchanges potential energy for kineticenergy. Some of that potential energy is transformed into translational kinetic energyand some is transformed into rotational kinetic energy. The manner in which that potential energy is distributed between the two forms of kinetic energy has animpact on how the object rolls.
Cylinders, disks, and spheres
Imagine a cylinder-like, disk-like, or sphere-like object that has a mass M and a radius R rolling down an incline. The moment of inertia for each of four differentgeometrical examples is shown in Figure 1 .
Figure 1 . Examples of moment of inertia. |
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Thin hollow cylindrical shape or hoop I = M*R^2 Solid cylinder or disk I = (1/2)*M*R^2 Solid sphere I = (2/5)*M*R^2 Then hollow spherical shell I = (2/3)*M*R^2 |
A very important constant
As you can see from Figure 1 , for these four shapes at least, the moment of inertia is equal to a constant multiplied by the product of the mass andthe square of the radius. (As far as I know, this constant hasn't been given a widely-recognized name.)
In these four cases, we can write an expression for the moment of inertia for rotation about the center of mass as
Icm = Q*M*(R^2)
where
- Icm represents the moment of inertia through the center of mass
- Q represents a constant that applies for a particular shape
- M represents the mass
- R represents the radius
The purpose of the constant
The purpose of the constant, Q, is to specify how the mass is distributed relative to the axis of rotation. Larger values of Q generally indicate that themass is distributed further from the axis of rotation. Similarly, larger values of Q also indicate larger moments of inertia.
Rotational versus translational kinetic energy
Given the relationships
Icm = Q*M*R^2, and
Vcm = W*R, and
Krot = (1/2)*Icm*W^2
where
- Icm represents the moment of inertia through the center of mass
- Q represents a constant that applies for a particular shape
- M represents the mass
- R represents the radius
- Vcm represents the translational velocity of the center of mass
- W represents the angular velocity
- R represents the radius
- Krot represents the rotational kinetic energy that you learned about in an earlier module .
We can substitute and write
Krot = (1/2)*(Q*M*R^2)*(Vcm/R)^2, or
Krot = (1/2)*(Q*M*R^2)*(Vcm^2/R^2), or
Krot = (1/2)*(Q*M)*(Vcm^2), or
Krot = Q*(1/2)*(M)*(Vcm^2)
Given that
Ktr = (1/2)*(M)*(Vcm^2)
where
- Ktr represents translational kinetic energy, which you also learned about in the earlier module