In the equation x = 6 sin(0.5πt), the motion is described by a sine function. The maximum amplitude of the motion is 6 units. To determine the time it takes for the dot to go from the middle (equilibrium position) to the maximum amplitude, we need to find the value of t that satisfies this condition.
The middle position is when x = 0. At the maximum amplitude, x = 6. Therefore, we can set up the equation:
6 sin(0.5πt) = 6
Dividing both sides by 6, we get:
sin(0.5πt) = 1
To find the time t, we need to find the angle whose sine is equal to 1. The angle with a sine of 1 is π/2 (90 degrees).
0.5πt = π/2
Simplifying, we have:
t = (π/2) / (0.5π)
t = 2
Therefore, the dot will take 2 units of time to go from the middle to the maximum amplitude in simple harmonic motion.