Question

Using a funnel and a marble or a ball bearing try to work ouf the situation in the above question. Try to realize that as the marble goes towards the brim, its linear speed increases but its angular speed decreases. When nearing the base, it is the other way.

Answer 1

The centripetal force condition gives,

N cos θ = `(m v^2)/r`

N sin θ = mg

∴ tan θ = `(r g)/v^2`

⇒ v = `sqrt((r g)/(tan theta))`

So, as the marble moves towards the brim, the radius r increases. Hence, its linear speed v increases.

v = rω

∴ ω = `v/r`

Since v ∝ `sqrt r`,

ω ∝ `1/sqrt r`

Hence, as the marble moves towards the brim, its angular speed decreases.

Similarly, when the marble moves towards the base, r decreases, so its linear speed decreases, but its angular speed increases.