# Pendulum max gravity acceleration

#### terryds

1. The problem statement, all variables and given/known data

What is the ratio between maximum acceleration of pendulum oscillation and the gravity acceleration ?
Express the answer in terms of L (the length of pendulum string)

2. Relevant equations
SHM

3. The attempt at a solution

amax = ω2 A = (g/l) L sin θ = g sin θ

So, the ratio is sin θ..
But, how to express sin θ in terms of L?
I know that for small angles, sin θ can be approximated to θ, and θ is arc length/L... Still, it's confusing

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#### haruspex

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The question strikes me as ambiguous. Does it mean the maximum angular acceleration, $\ddot \theta$, or the maximum linear tangential acceleration, $L\ddot \theta$?
If we take it as linear, dimensional analysis shows the question is unanswerable. A ratio of two accelerations is dimensionless, so cannot be derived from a single distance. At least two distances would be required.
If we take it as angular, we still don't get any further since, as you found, it depends on the amplitude.

Another possibility is total linear acceleration, which means centripetal acceleration needs to be considered.

#### terryds

The question strikes me as ambiguous. Does it mean the maximum angular acceleration, $\ddot \theta$, or the maximum linear tangential acceleration, $L\ddot \theta$?
If we take it as linear, dimensional analysis shows the question is unanswerable. A ratio of two accelerations is dimensionless, so cannot be derived from a single distance. At least two distances would be required.
If we take it as angular, we still don't get any further since, as you found, it depends on the amplitude.

Another possibility is total linear acceleration, which means centripetal acceleration needs to be considered.
The options are

A. 2L
B. √L
C. √(1/L)
D. L
E. 1/L

Which one is correct... please help

#### haruspex

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The options are

A. 2L
B. √L
C. √(1/L)
D. L
E. 1/L

Which one is correct... please help
I would say this establishes that what they are after is angular acceleration, $\ddot \theta$, not linear acceleration. We still have the problem that the correct answer involves the amplitude, but maybe the question intended to ask only how the ratio depends on L, rather than an exact ratio between the two accelerations, so just treat it as though the amplitude is 1.
But there is a second difficulty. The "ratio between" does not specify which is to be divided by the other. Is a ratio of 1:L an answer of L or 1/L?

• terryds

#### terryds

I would say this establishes that what they are after is angular acceleration, $\ddot \theta$, not linear acceleration. We still have the problem that the correct answer involves the amplitude, but maybe the question intended to ask only how the ratio depends on L, rather than an exact ratio between the two accelerations, so just treat it as though the amplitude is 1.
But there is a second difficulty. The "ratio between" does not specify which is to be divided by the other. Is a ratio of 1:L an answer of L or 1/L?
It means the division of max pendulum acceleration by the gravitational acceleration. Maybe it's 1/L

#### haruspex

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It means the division of max pendulum acceleration by the gravitational acceleration. Maybe it's 1/L
That looks the most likely.

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