We are given that a mass is attached to a string and is spun is in a horizontal circle, this means that the free body diagram of the problem is the following:
We will add the forces in the vertical direction:
The forces add up to zero since there is no acceleration in the vertical direction. Therefore, we can add "mg" to both sides:
Now, The vertical component of the tension can be put in terms of the total tension using the following right triangle:
Therefore, we use the trigonometric function sine:
Multiplying both sides by "T"
Substituting in the sum of vertical forces:
Now, Since the horizontal component of the tension is equivalent to the centripetal force, we have that:
Where:
We can use the trigonometric function cosine to determine the horizontal component of the tension:
Multiplying both sides by "T":
Substituting we get:
Now, we divide both equations:
Simplifying we get;
Now, we multiply both sides by "g":
Now, in any circular motion, the period "P" is given by:
Where:
Also, the acceleration is given by:
Substituting the expression for the acceleration we determined before we get:
Substituting the expression for the period:
Solving the square:
Solving the fraction:
Simplifying:
Now, we can put the radius in terms of the length of the spring using the following triangle:
using the function sine we get:
Substituting in the previous equation we get:
Now, we decompose the tangent:
Simplifying:
Now, we invert both sides:
Multiplying both sides by "g" we get:
The angular velocity is the angle divided by time, since it takes 0.644 s to complete one revolution this means that :
substituting the values:
Solving the operations:
Now, going back to the sum of forces in the vertical direction:
Dividing both sides by "g":
Substituting the values:
Solving the operations:
Therefore, the mass of the ball is 0.021 kg.