Brothers Meets Physics

I was playing a game called “Brothers” when I noticed a sweet physics problem. Check out the video below:

My question was: is it possible for the little brother to hold on while his big brother makes those crazy swings? Here is what I found.

First, we should make some assumptions about the lengths involved. It looks to me like the rope is about 3 times as long as the big brother. Assuming the big brother is 5 feet tall (they are young boys), this gives us a rope length of 4.5m (see below).

physics problem 2

To determine the force exerted on the little brother, we have the following factors. First, we have the weight of the big brother. Second, we have the centripetal motion of the big brother. As a formula, it would look like this:

F = F_{g} + F_{c}

F = mg + \frac{mv^{2}}{r}

We know the radius of motion is the length of the rope. We won’t specify the mass of the big brother right now. Hence, we need to determine the velocity of the big brother. To do this, we can use energy considerations.

When the big brother is grabbing onto the wall, all of his energy is gravitational potential energy. When he is at the bottom of his swing, all of his energy is kinetic energy.

E_{g} = E_{k}

mgh = \frac{1}{2} mv^{2}

Rearranging the fomula above for v2 gives:

v^{2} = 2gh

Substituting this info our equation for force, we have:

F = mg + \frac{m2gh}{r}

However, h (the height of the drop) and r (the radius of the swing) are the same. Therefore, our formula reduces to:

F = mg + 2mg

F = 3mg

Since we solved the above algebraically, we can be certain that no matter the length of the rope, the force on the little brother will be 3 times the weight of the big brother. Could the little brother hold 3 copies of his big brother dangling from a rope? I don’t know. It seems plausible. Regardless, the physics is quite interesting.

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