A ball on the end of a string is revolving at a uniform rate in a vertical circle of radius 98.8 cm. If its speed is 4.60 m/s, and its mass is 0.345 kg, calculate the tension (in newtons) in the string when the ball is at the bottom of the path.
To calculate the tension in the string when the ball is at the bottom of the path, we can use the centripetal force equation.
The centripetal force is given by the formula: F = (m * v^2) / r
Where:
F is the centripetal force,
m is the mass of the ball,
v is the speed of the ball,
and r is the radius of the circle.
Given that:
m = 0.345 kg,
v = 4.60 m/s,
and r = 98.8 cm = 0.988 m,
We can plug these values into the formula and solve for F:
F = (0.345 kg * (4.60 m/s)^2) / 0.988 m
First, let's calculate (4.60 m/s)^2:
(4.60 m/s)^2 = 21.16 m^2/s^2
Next, divide this value by 0.988 m:
F = (0.345 kg * 21.16 m^2/s^2) / 0.988 m
F = 7.4702 kg * m/s^2
So, the tension in the string when the ball is at the bottom of the path is approximately 7.4702 N (newtons).
To calculate the tension in the string when the ball is at the bottom of the path, we can use the principles of centripetal force.
The tension in the string provides the centripetal force required to keep the ball moving in a circular path. At the bottom of the path, the tension in the string is at its maximum, as it must counteract the downward force due to the ball's weight.
The centripetal force required to keep the ball moving in a circle is given by the equation:
F = (m * v^2) / r
Where:
F is the centripetal force
m is the mass of the ball
v is the velocity of the ball
r is the radius of the circular path
We are given:
m = 0.345 kg (mass of the ball)
v = 4.60 m/s (velocity of the ball)
r = 98.8 cm = 0.988 m (radius of the circular path)
Using the given values, we can substitute them into the equation to find the centripetal force:
F = (0.345 kg * (4.60 m/s)^2) / 0.988 m
Now, let's calculate the tension in the string:
F = (0.345 kg * 21.16 m^2/s^2) / 0.988 m
F = 7.58648 kg·m/s^2 / 0.988 m
F ≈ 7.68 N
Therefore, the tension in the string when the ball is at the bottom of the path is approximately 7.68 Newtons.