Final answer:
To ensure the chain goes slack at the top of the rotation, Timmy should twirl the rock at a velocity of approximately 3.13 m/s, calculated using the relationship between centripetal force and gravitational force.
Step-by-step explanation:
The question deals with the concept of circular motion in physics. Specifically, we need to find the velocity at which Timmy should rotate the rock so that the chain is just about to go slack at the top of the circular path. This occurs when the centripetal force required for circular motion equals the gravitational force acting on the rock, meaning that the tension in the chain is zero.
At the top of the circular path, the only forces acting on the rock are gravity and the tension in the chain. The tension is zero when the chain goes slack. Thus, the centripetal force needed to keep the rock in circular motion is provided entirely by the weight of the rock (mg), where m is the mass of the rock and g is the acceleration due to gravity.
The centripetal force Fc is given by the equation Fc = mv2/r, where m is the mass of the rock, v is the velocity, and r is the radius of the circle. Setting the centripetal force equal to the gravitational force (mg) and solving for v, we get v = √(gr).
Substituting the given values m = 2.0 kg and r = 1.0 m, and assuming g = 9.8 m/s2, we calculate the required velocity. Thus:
v = √(9.8 m/s2 × 1.0 m)
v = 3.13 m/s (approximately)
Timmy should rotate the rock at a velocity of approximately 3.13 m/s for the chain to go slack at the top of the circle.