A ball on the end of a string is cleverly revolved at a uniform rate in a vertical circle of
radius 85.0cm. If its speed is 4.15 m/s and its mass is 0.300 kg, calculate the tension in the
string when the ball is (a) at the top of its path, and (b) at the bottom of its path.
To calculate the tension in the string, we can use the centripetal force equation:
Centripetal force (F) = (mass (m) * velocity^2 (v^2)) / radius (r)
(a) At the top of the ball's path:
In this case, the tension in the string will be the sum of the centripetal force and the weight of the ball. The weight can be calculated using the formula: Weight (W) = mass (m) * acceleration due to gravity (g).
First, let's calculate the weight of the ball:
Weight = m * g = 0.300 kg * 9.8 m/s^2 = 2.94 N
Now, let's calculate the centripetal force:
F = (m * v^2) / r = (0.300 kg * (4.15 m/s)^2) / 85.0 cm
Note: We need to convert the radius from centimeters (cm) to meters (m).
1 meter = 100 centimeters, so 85.0 cm = 85.0 / 100 = 0.85 m
F = (0.300 kg * (4.15 m/s)^2) / 0.85 m = 6.66 N
At the top of the path, the tension in the string is the sum of the centripetal force and the weight:
Tension = F + Weight = 6.66 N + 2.94 N = 9.60 N
Therefore, the tension in the string at the top of the ball's path is 9.60 Newtons.
(b) At the bottom of the ball's path:
At the bottom of the ball's path, the tension in the string will be less than at the top since the weight of the ball assists with the centripetal force.
Again, we need to calculate the weight of the ball:
Weight = m * g = 0.300 kg * 9.8 m/s^2 = 2.94 N
Since the weight assists with the centripetal force, we subtract it from the centripetal force:
Tension = F - Weight
Using the same formula for centripetal force as before:
F = (m * v^2) / r
Tension = F - Weight = ((m * v^2) / r) - Weight
Substituting the given values:
Tension = ((0.300 kg * (4.15 m/s)^2) / 0.85 m) - 2.94 N
Calculating the tension in the string at the bottom of the ball's path will give you the answer.