# Balancing a plank on a cylinder

• simon janssens
In summary, the problem involves finding the period of small oscillations for a uniform thin rigid plank rolling without slipping on top of a rough circular log. The resulting force is always perpendicular to the plank's center of mass, resulting in a torque of mgaϑ. The moment of inertia is calculated using the formula for parallel axes of rotation, and the angular acceleration is found to be ϑ'' = 3ga/b2 ⋅ ϑ. However, a mistake in calculating the torque results in the angular acceleration being opposed to ϑ. After correcting the mistake, the period is found to be T = 2πb√(ϑ0/(3ga)). However, the calculation for ω may be incorrect and requires
simon janssens

## Homework Statement

The figure shows a uniform thin rigid plank of length 2b which can roll
without slipping on top of a rough circular log of radius a. The plank is initially
in equilibrium, resting symmetrically on top of the log, when it is slightly
disturbed. Find the period of small oscillations of the plank.

The wording of the question implicates that we can use sinϑ ≈ ϑ and cosϑ ≈ 1 because ϑ2 ≈ 0.
The plank has no thickness and the mass is uniformly distributed.
Gravity is constant.

## The Attempt at a Solution

I figured that the resulting force is always pointed perpendicular to the plank in the center of mass. This results in a torque τ = r × F .
r = |GC| = ϑa because there is no slipping
|F| = mgcosϑ = mg (rounding for small angles)
so the torque is τ = mgaϑ

Then I try calculating the moment of inertia of the plank, I use the formula for parallel axes of rotation so that the moment of inertia is I = ICM + mh2
ICM = ml2/12 = mb2/3
mh2 = m|GC|2 = ma2ϑ2 = 0
so the moment of inertia is I = mb2/3

This gives me enough to calculate the angular acceleration, ϑ'' = τ/I = 3ga/b2 ⋅ ϑ
I deduce that ϑ is directly proportional with its second derivative.
ϑ has to be of the form eλt or Asin(ωt +φ0).
Because I'm looking for an oscillation, my guess is that it will be of the latter.
Now I have 3 variables and 3 equations, at t = 0 we get :
ϑ(0) = ϑ0 = Asin(ω⋅0 +φ0)
ϑ'(0) = 0 = ωAcos(ω⋅0 +φ0)
ϑ''(0) = 3ga/b2 ⋅ ϑ0 = -ω2Asin(ω⋅0 +φ0)

I tried solving these, but it didn't work, can you help me please ?
Thanks a lot,
S.

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simon janssens said:
This gives me enough to calculate the angular acceleration, ϑ'' = τ/I = 3ga/b2 ⋅ ϑ
.
When θ is positive, will its acceleration be positive or negative?

haruspex said:
When θ is positive, will its acceleration be positive or negative?
It should be negative, because the plank is always accelerating towards the point of equilibrium. So should it be ϑ''(0) = -3ga/b2 ⋅ ϑ ?
I probably made a mistake with the vector product τ = r × F

UPDATE ----
thanks to haruspex I noticed that i calculated τ = r × F incorrectly, when taking clockwise as positive, the torque should be negative.
This makes the angular acceleration opposed to ϑ, as expected.
this gives 3 equations and 3 variables, as before :
ϑ(0) = ϑ0 = Asin(φ0)
ϑ'(0) = 0 = ωAcos(φ0)
ϑ''(0) = -3ga/b2 ⋅ ϑ0 = -ω2Asin(φ0) = -ω2ϑ0

this solves to
φ0 = π/2
A = ϑ0
ω = 1/b ⋅√(3ga/ϑ0)
the period of this function is T = 2πb√(ϑ0/(3ga))

is this correct ? can someone check please ?

simon janssens said:
ω = 1/b ⋅√(3ga/ϑ0)
I would not expect ω to depend on the amplitude. Please check your working.

## 1. How does balancing a plank on a cylinder work?

When a plank is balanced on a cylinder, the center of gravity of the plank must be directly above the center of the cylinder. This creates stability and prevents the plank from falling over.

## 2. What is the physics behind balancing a plank on a cylinder?

Balancing a plank on a cylinder involves the principles of equilibrium and center of mass. The center of mass of the combined system (plank and cylinder) must remain within the base of support (the surface area defined by the cylinder) in order for the plank to remain balanced.

## 3. What factors affect the stability of balancing a plank on a cylinder?

The stability of balancing a plank on a cylinder is affected by several factors, including the diameter of the cylinder, the length and width of the plank, the weight of the plank, and the position of the center of gravity of the plank.

## 4. Why is balancing a plank on a cylinder a common science experiment?

Balancing a plank on a cylinder is a common science experiment because it demonstrates important principles of physics, such as equilibrium, center of mass, and stability. It also allows for hands-on exploration and observation of these concepts.

## 5. What are some real-world applications of balancing a plank on a cylinder?

The concept of balancing a plank on a cylinder has real-world applications in engineering and architecture. It is also used in designing structures such as bridges and towers, where the center of mass must be carefully considered to ensure stability and prevent collapse.

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