A bucket of mass 2.20 kg is whirled in a vertical circle of radius 1.15 m. At the lowest point of its motion the tension in the rope supporting the bucket is 28.0 N. a) Find the speed of the bucket.
b)How fast must the bucket move at the top of the circle so that the rope does not go slack?
a.) 1.65 m/s
To find the speed of the bucket at the lowest point of its motion, we can use the principle of conservation of energy.
a) At the lowest point, all of the gravitational potential energy is converted into kinetic energy. The formula for gravitational potential energy is given by:
PE = m*g*h
Where m is the mass of the bucket, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the bottommost point.
Since the height at the lowest point is equal to the radius of the circle (1.15 m), we can calculate the gravitational potential energy:
PE = 2.20 kg * 9.8 m/s^2 * 1.15 m = 25.919 J
Since all of the potential energy is converted into kinetic energy at the lowest point, we can equate the two:
KE = 1/2 * m * v^2
Where KE is the kinetic energy of the bucket and v is its speed at the lowest point.
Solving for v, we have:
v = sqrt(2 * KE / m)
Substituting the known values:
v = sqrt(2 * 25.919 J / 2.20 kg)
v = sqrt(23.562 m^2/s^2) ≈ 4.854 m/s
Therefore, the speed of the bucket at the lowest point is approximately 4.854 m/s.
b) To determine the required speed at the top of the circle, we need to consider the tension in the rope. At the topmost point, the tension in the rope is equal to the sum of the gravitational force and the centripetal force. The formula for the tension is given by:
T = m * g + m * v^2 / r
Where T is the tension, m is the mass, g is the acceleration due to gravity, v is the speed, and r is the radius of the circle.
We know that the tension at the lowest point is given as 28.0 N. Therefore, we can set up the equation:
T = m * g + m * v^2 / r
28.0 N = 2.20 kg * 9.8 m/s^2 + 2.20 kg * v^2 / 1.15 m
Let's solve this equation to find the required speed v at the top of the circle:
2.20 kg * v^2 / 1.15 m = 28.0 N - 2.20 kg * 9.8 m/s^2
v^2 = (28.0 N - 2.20 kg * 9.8 m/s^2) * 1.15 m / 2.20 kg
v^2 = 65.8 N*m - 22.0 N*m
v^2 = 43.8 N*m
v = sqrt(43.8 N*m) ≈ 6.621 m/s
Therefore, the bucket must move at a speed of approximately 6.621 m/s at the top of the circle to ensure that the rope does not go slack.