A bucket of mass 2.20 kg is whirled in a vertical circle of radius 1.40 m. At the lowest point of its motion the tension in the rope supporting the bucket is 30.0 N. Find the speed of the bucket. How fast must the bucket move at the top of the circle so that the rope does not go slack?
To find the speed of the bucket, we can use the concept of centripetal force.
First, let's consider the forces acting on the bucket at the bottommost point of the circle. The weight of the bucket acts downward (mg), and the tension in the rope acts upwards (T). At this point, the net force acting on the bucket is the difference between these two forces:
Net force = T - mg
Since the bucket is moving in a circle, the net force acting on it must provide the centripetal force:
Net force = m * (v^2 / r)
Here, m is the mass of the bucket, v is the speed of the bucket, and r is the radius of the circle.
Setting the two equations equal to each other, we have:
T - mg = m * (v^2 / r)
Now, we can substitute the given values into the equation. The mass of the bucket (m) is 2.20 kg, the radius of the circle (r) is 1.40 m, and the tension in the rope (T) is 30.0 N.
30.0 N - (2.20 kg * 9.8 m/s^2) = 2.20 kg * (v^2 / 1.40 m)
Solving for v^2, we have:
v^2 = [(30.0 N - 2.20 kg * 9.8 m/s^2) * 1.40 m] / 2.20 kg
Calculating this expression gives:
v^2 ≈ 111.28 m^2/s^2
To find the speed of the bucket, we can take the square root of both sides:
v ≈ √(111.28 m^2/s^2)
v ≈ 10.55 m/s
Therefore, the speed of the bucket is approximately 10.55 m/s when it is at the lowest point of its motion.
To find the required speed at the top of the circle so that the rope does not go slack, we need to determine the minimum tension in the rope at that point.
At the topmost point of the circle, the forces acting on the bucket are the weight of the bucket (mg) acting downward and the tension in the rope (T) acting downward. The net force at this point should provide the centripetal force, directed towards the center of the circle.
Net force = T + mg
Since the rope should not go slack, the net force cannot be zero. Therefore, we find the minimum tension (Tmin) required for this condition:
Tmin + mg > 0
Tmin > -mg
Now, we can substitute the values into the equation. The mass of the bucket (m) is 2.20 kg, and the acceleration due to gravity (g) is approximately 9.8 m/s^2.
Tmin > -(2.20 kg * 9.8 m/s^2)
Tmin > -21.56 N
Therefore, the tension in the rope at the top of the circle should be greater than 21.56 N, so that it does not go slack.