A bucket of mass 1.60 kg is whirled in a vertical circle of radius 1.30 m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket.
Enter a number.1 m/s
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?
Enter a number.2 m/s
the .1 and .2 or like blanks where the number should be
at the lowest point, tension=mg+mv^2/r
at the top, tension=mv^2/r-mg
for it to go slack, tension would be zero.
To solve this problem, we can use the principles of centripetal force and gravitational force.
(a) To find the speed of the bucket at the lowest point of its motion, we need to consider the tension in the rope and the gravitational force acting on the bucket. At the lowest point, the tension in the rope must provide the centripetal force to keep the bucket moving in a circular path.
The centripetal force (Fc) is given by the formula:
Fc = (mass * velocity^2) / radius
where mass is the mass of the bucket (1.60 kg), velocity is the speed of the bucket, and radius is the radius of the circle (1.30 m).
The gravitational force (Fg) acting on the bucket is given by the formula:
Fg = mass * gravity
where gravity is the acceleration due to gravity (9.8 m/s^2).
Since the tension in the rope is equal to the sum of the centripetal force and gravitational force:
Tension = Fc + Fg
Substituting the known values:
25 N = ((1.60 kg) * velocity^2) / (1.30 m) + (1.60 kg * 9.8 m/s^2)
Simplifying the equation and solving for velocity:
25 N = (1.23 kg * velocity^2) + (15.68 kg * m/s^2)
Simplifying further:
25 N = 1.23 kg * velocity^2 + 15.68 N
Rearranging the equation:
1.23 kg * velocity^2 = 25 N - 15.68 N
1.23 kg * velocity^2 = 9.32 N
Dividing both sides of the equation by 1.23 kg:
velocity^2 = 7.5854 m^2/s^2
Therefore, the speed of the bucket at the lowest point of its motion is:
velocity = sqrt(7.5854 m^2/s^2)
So, the correct answer is approximately 2.75 m/s.
(b) Now, we need to find the minimum speed required at the top of the circle so that the rope does not go slack. At the top of the circle, the tension in the rope must be equal to the gravitational force acting on the bucket.
Using the same formulas as before, we can set up the following equation:
Tension = Fc + Fg
Tension = (mass * velocity^2) / radius + mass * gravity
Substituting the known values:
25 N = ((1.60 kg) * velocity^2) / (1.30 m) + (1.60 kg * 9.8 m/s^2)
Solving for velocity:
25 N = (1.23 kg * velocity^2) + (15.68 kg * m/s^2)
1.23 kg * velocity^2 = 25 N - 15.68 N
1.23 kg * velocity^2 = 9.32 N
Dividing both sides of the equation by 1.23 kg:
velocity^2 = 7.5854 m^2/s^2
Therefore, the minimum speed required at the top of the circle is:
velocity = sqrt(7.5854 m^2/s^2)
So, the correct answer is approximately 2.75 m/s.