Karate

Classical Mechanics Level 1

Is it easier for the person on the left to break a uniform wooden board of length $l$ , which has a mass $m$ than to break a uniform wooden board of twice the length $2l$ and twice the mass $2m$ ?

Note : Both are equivalent in thickness, density, etc. The person doing the kicking contacts the center of the boards in both cases. Assume that, if the board breaks, it breaks almost immediately after the collision.

No Yes

7 solutions

Jake Lai
Jul 26, 2015

It might be easy to follow $F = ma$ and conclude that a longer board would be harder, but one has to consider torque instead. Since the pivot arm is twice as long, breaking a doubly long board would be around twice as easy.

Try breaking a chopstick in half, and then into a halfpiece's half. Which seems easier? (Apply the force on the center.)

I need clarification. The question mentions that the board is lengthened to twice it's length. When boards are measured the length is always measured with the grain of the wood, and the width is measured across the grain. If the question would have said the width of the board was changed, my answer would have been different. I still believe a longer board in grain direction would be more difficult to break than increasing the dimension in the width direction.

Brian Heiland - 5 years, 10 months ago

While it's possible that wood texture and grain adds an additional variable, its a simple torque problem, each hand is providing a force in the opposite direction of the kick. The further each force gets from the point of impact, the less force (via torque) it applies in opposition of the kick. Since mass is uniformly proportional between the 2 boards it becomes irrelevant.

Matthew Kirkeby - 5 years, 10 months ago

With twice the length for a fixed density and thickness, the board is effectively twice in mass.

The impact point being in the center of said board, doubling the lenght also doubles the lever arm. For a fixed amount of force, that translates to double the torque.

The material being the same, same density and same thickness, this should make it easier to break.

Sean Diano - 5 years, 10 months ago

I am a little confused on this one, intuitively speaking yes your chopstick example makes sense but if I keep on increasing the size of the board to say 10l or 20l with the mass multiplying proportionately then it doesn't seem as easy.

Amit Nihal - 5 years, 10 months ago

Couldn't we use the equation of torque... (vector) tau=(vector) r × (vector) F???

Istiak Reza - 5 years, 10 months ago

But it would be harder for the holder of the wood to resist the blow for the very same reason.

Mark Smart - 5 years, 10 months ago

I get the chopstick theory but could you please break the torque and pivot arm thing down? I am confused

Nehemiah Osei - 5 years, 10 months ago

I think there should be an optimal length of the board, where the required force and required displacement are both feasible for a person. If the beam is too short, it takes too much force to break, but if the beam is too long, it requires too much displacement to break (it simply bends). The optimal length of the beam is analogous to the optimal gearing on a bicycle.

Sean Lourette - 3 years, 8 months ago

Would the mass of the two boards being different have any effect?

Aviral Jain - 5 years, 10 months ago

If the man kick the board with F. There are the torque will be produce. Therefore, in the board which have mass m and length has T=F l/2 here we subjected the F , F= 2T/l. But in the board which has mass 2m and length 2l it has T=F 2l/2 . By subjecting F , F=T/l. From this conclusions higher force need to break the board with mass m and length l.

SanuJan Uthaya - 4 years, 9 months ago

ชาญชนะ สอนสุนทร
Aug 6, 2015

Assume the board's both ends are pivot points

From M = F * L

M = Moment, the power you can make something rotate

F = Force, your foot!

L = Length from the center to the pivot point

You kick it in the center is like you making the board rotate, M is high if length is long, length can be adjusted by using longer board.

So the longer board is easier to break because you can managed to easily rotate it in the center.

Thom Verga
Aug 9, 2015

Wood breaks most easily going with the grain. If they had said 2x the width the answer would have been yes because you would have increased the length significantly in the direction it is most likely to break .

Vikram Singh
Apr 21, 2017

It's all about bending stress induced in the plank. Bending stress= M*Y/I . where M(max)= FL/2 for centrally loaded simply supported beam. Thus as length of the board increases the bending moment 'M' increases which inturn results in increased bending stress in the centre of board and so it is easy to break a long board.

Hari Om Sharma
Aug 12, 2015

We can use F=ma

How would tell the F= ma can use

SanuJan Uthaya - 4 years, 9 months ago

using relations in momentum and force

Hari Om Sharma - 4 years, 9 months ago

Andriane Casuga
Aug 10, 2015

the second one , if we set the two as for example a simple beam (2 supports at both ends then apply the load at the center) the kind of stress is flexure, sigma=Mc/I , the c(outermost fiber from the neutral axis) is the same in the 2 materials as well as the I(rectangular moment of inertia) but the M(maximum bending moment) is different. M=PL/4 in a simple beam so imagine if the L is doubled, the M will also doubled so, therefore the induced stress is higher in the second wood than the first. smile emoticon sorry for my bad english

Bhooshan Deshpande
Aug 6, 2015

Board is most likely broken by bending force in both cases rather than other types of failure. So bending strength must be considerd of boards. Bending strength is proportional to applied moment which is larger in case of board of length 2l for same force. So , second board will break easily than the first one. Refer flexural formula if required.

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