HBO is concerned about injuries on the set of their award winning fantasy series Game of Thrones. Star characters are given real swords which have a point of balance (centre of mass) 5.0cm above the top of the handle. For safety reasons, you are given only the 15cm long, 0.25kg uniform handle and are told to modify it such that the point of balance is shifted to top of the handle. You are told that you are only allowed to add a dense mass to the bottom of the handle and that the blade component is uniform and 90cm long. To assemble the sword, the blade is inserted into the handle and spans the full length. A guard separating the exposed portion of the blade and the handle is also present; length-wise, it is negligibly thin but weighs 0.50kg. How much mass do you add?

Question

HBO is concerned about injuries on the set of their award winning fantasy series Game of Thrones. Star characters are given real swords which have a point of balance (centre of mass) 5.0cm above the top of the handle. For safety reasons, you are given only the 15cm long, 0.25kg uniform handle and are told to modify it such that the point of balance is shifted to top of the handle. You are told that you are only allowed to add a dense mass to the bottom of the handle and that the blade component is uniform and 90cm long. To assemble the sword, the blade is inserted into the handle and spans the full length. A guard separating the exposed portion of the blade and the handle is also present; length-wise, it is negligibly thin but weighs 0.50kg. How much mass do you add?

I know I have to use the centre of mass equation somehow...

Answer 1

To solve this problem, you can use the concept of the center of mass and leverage the equation for calculating it. The equation you need to use is:

M1 x D1 = M2 x D2

Where:
M1 and M2 are the masses on either side of the fulcrum (in this case, the point of balance).
D1 and D2 are the distances of the masses from the fulcrum.

In this scenario, the handle, guard, and blade can be considered as a system with two masses: the handle with an added mass at the bottom, and the blade with the guard.

Let's set up the equation for the handle system:
(M_handle + M_added_mass) x D_handle = M_blade x D_blade

Given information:
M_handle = 0.25 kg (mass of the 15cm handle)
D_handle = 5.0 cm (distance from the point of balance to the top of the handle)
M_blade = 0.90 kg (mass of the 90cm blade)
D_blade = 90 cm (distance from the point of balance to the top of the blade)

We need to solve for M_added_mass, which represents the additional mass that needs to be added to the handle.

First, convert the distances to meters:
D_handle = 0.05 m
D_blade = 0.90 m

Now, substitute the given values into the equation:
(0.25 + M_added_mass) x 0.05 = 0.90 x 0.90

Solve the equation to find the value of M_added_mass:
0.0125 + 0.05M_added_mass = 0.81
0.05M_added_mass = 0.81 - 0.0125
0.05M_added_mass = 0.7975
M_added_mass = 0.7975 / 0.05
M_added_mass ≈ 15.95 kg

Therefore, to shift the point of balance to the top of the handle, you would need to add approximately 15.95 kg of mass to the bottom of the handle.