Final answer:
Using conservation of energy principles, the mass attached to the wheel will rise to a maximum height of approximately 0.681 meters above point P when the wheel is released.
Step-by-step explanation:
The question concerns a physical scenario where a wheel of radius 20.0 cm and mass 2.90 kg is rotating at a constant rate, raising a mass m = 0.785 kg until it reaches a point P with an upward speed of 3.65 m/s. Once the wheel is released, it starts to slow down, eventually reversing its direction due to the downward tension of the cord. To find the maximum height h the mass will rise above point P, we can use the principle of conservation of energy; specifically, the transformation of kinetic energy into gravitational potential energy.
At the moment when the wheel is released, the kinetic energy can be expressed as K.E. = 0.5 * m * v^2, where m is the mass of the weight and v is its speed at point P. As the wheel slows down, this kinetic energy is converted to gravitational potential energy, which is given by P.E. = m * g * h, with g being the acceleration due to gravity (9.8 m/s^2), and h the height above point P.
Equating the kinetic energy at point P with the potential energy at the maximum height, we get:
0.5 * m * v^2 = m * g * h
Solving for h, we find:
h = (0.5 * v^2) / g
Inserting the given values:
h = (0.5 * (3.65 m/s)^2) / (9.8 m/s^2)
Calculating gives us:
h = 0.681 m
Therefore, the maximum height the mass m will rise above the point P is approximately 0.681 meters.