Three objects of uniform density- a solid sphere, a solid cylinder, and a hollow cylinder- are placed at the top of an incline. They are all released from rest at the same elevation and roll without slipping. Which object reaches the bottom first? Which reaches it last? Try this at home and note that the result is independent of the masses and the radii of the objects.
Anonymous
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momen inertia of (each object is solid)
I_sphere = (2/5)mr²
I_cylinder = (1/2)mr²
I_thin_cylinder = mr²
torque = I α
(friction) r = I (a/r)
friction = Ia/r²
second Newton's law of linear motion,
ΣF = m a
m g sin θ - friction = m a
m g sin θ - Ia/r² = m a
m g sin θ = (m + I/r²)a
a = m g sin θ/(m + I/r²)
a ∼ 1/I
acceleration inversely proportional with respect to the moment of inertia,
I_sphere < I_cylinder < I_thin_cylinder
The sphere reaches the bottom for the first time and thin_cylinder for the last one.
'Try this at home and note that the result is independent of the masses and the radii of the objects.'
moment of inertia I = kmr²
hence k is a real constant.
a = m g sin θ/(m + I/r²)
a = m g sin θ/(m + kmr²/r²)
a = g sin θ/(1 + k)
acceleration a is independent of the masses and the radii of the objects.
Physicsquest
1st = solid sphere
2nd = solid cylinder
3rd = hollow cylinder