Rotational Inertia Stick Demosaxbyphysics.com/AdvancedPhysicsLab/Mechanics/Adv... · Web viewR with...
Transcript of Rotational Inertia Stick Demosaxbyphysics.com/AdvancedPhysicsLab/Mechanics/Adv... · Web viewR with...
Rolling Down an Incline 1. Once upon a time, we showed that an object rolling down an incline of angle θ (from the horizontal) would have acceleration a = g sin θ. Strictly speaking, this isn’t quite true. Now that we know a bit about rotational inertia, let’s find the actual acceleration a of a radially symmetric object (could be a disc, cylinder or sphere, and could be hollow or solid) of mass m and radius R with rotational inertia I around the center of mass.
Putting it all together…and solving for a:
Aside: Show that the acceleration you found above gives the expected result (a = g sin θ) when we ignore rotational inertia: that is, let I → 0.
Find vf for the object rolling down an incline of height h starting from rest, using an your expression for a boxed above (which includes rotational effects) and an equation of motion. (Get vf in terms of m, h, I and R.)
Aside: What do you expect the expression for vf to simplify to, in the absence of rotational effects? Does it?
Draw all the forces acting on the object.Translational Dynamics: Start by applying Newt’s 2nd Law:
Notice that you have 2 unknowns, fs and a. Use rotational dynamics to get another equation involving fs and a. Apply the torque equation around the center of mass…Solve for fs in terms of I and a.
a =
vf =
fs =
m, IR
θ
1
2. OK, now let’s find vf (including rotational effects) the easy way. Use the energy approach to find vf (again starting from rest) in terms of m, h, I and R. Don’t forget to label the diagram, including the reference level. Include rotational kinetic energy.
Which way do you like better, the forces or the energy approach?
m, I
h
R
θ
2