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You have a balance beam, the kind of scale that tips from one side to the other, depending on the weight on each side. On each side is a beaker, half-filled with water. The sides are in balance. Now, on the left side, you submerge a ping-pong ball suspended by a string. On the right side, you submerge a steel ball of the same volume as the ping-pong ball suspended from a crane.
Does the balance beam tip to the right, to the left, or does it remain unchanged?
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Working on a symbolic problem dealing with translational kinematics in 1 dimension. Not sure I've done it right.
"At time t=0, a student throws a set of keys vertically upward to her sorority sister, who is in a window at distance h above. The second student catches the keys at time t. With what initial velocity were the keys thrown? What was the velocity of the keys just before they were caught?"
Being thrown straight up, the keys are affected by gravity, so we can use a particle under constant acceleration model to analyze this.
x(f) = x(i) + 1/2( V(i) + V(f) ) * t
where x(f) is the the final position, x(i) is the initial position, V(i) is the initial velocity, V(f) is the final velocity and t is the time.
h = 1/2( V(i) + V(f) ) * t,
2h/t = V(i) + V(f),
V(i) = 2h/t - V(f),
V(f) = 2h/t - V(i).
V(i) = ( 2h/t - V(i) ) - a*t,
2*V(i) = 2h/t - 9.8m/s^2 * t,
V(i) = h/t - 4.9m/s^2 * t,
Which is the answer to the first question. For second question,
V(f) = 2h/t - (h/t - 4.9m/s^2 * t)
V(f) = h/t + 4.9m/s^2 * t
That's the anwers I've arrived at. Any way I could refine these answers? Or have I made a mistake?
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Fuck. Looks like we have to rethink everything again.
>The amplituhedron is a theoretical geometric object in infinite dimensional space that dramatically simplifies calculations of particle interactions and challenges the notion that space and time are fundamental components of reality.
>When the volume of the amplituhedron is calculated in N=4 supersymmetric Yang-Mills theory, it describes the scattering patterns of subatomic particles. This suggests the possibility that the nature of the universe, both classical relativistic spacetime and quantum mechanics, can be described with simple geometry. Calculations can be done without quantum mechanic's properties of locality and unitarity, which could help investigate quantum gravity theory.
It also vastly simplifies the calculations of particle interactions, such that interactions which would require hundreds of Feynman diagrams and thousands of pages of calculations can be computed with a single equation.
Since the N=4 supersymmetric Yang-Mills theory is a toy theory that does not describe the real world, the relevance of this theory to the real world is currently unknown, but it provides promising directions for research into theories about the real world.