As shown in the picture to the right, a ball is suspended from the ceiling by a string of negligible mass. When the ball is set into motion such that it moves in a horizontal circle at a constant speed of 1.6m/s, the tension in the string is 7.7N. What is the length of the string?

Does the missing figure show the mass of the ball?

There is centripetal force of magnitude mv²/r (horizontal) and the weight (vertical) of mg to form the resultant tension of the string of 7.7N.
r is the horizontal radius, so must be adjusted for the angle the string makes with the vertical.

To find the length of the string, we can use the concept of centripetal force. When an object moves in a circular path with constant speed, there must be a force acting towards the center of the circle, called the centripetal force. In this case, the tension in the string provides the centripetal force.

The formula to calculate the centripetal force is:

F = (m * v^2) / r

Where:
F is the centripetal force,
m is the mass of the object,
v is the velocity of the object, and
r is the radius of the circular path.

In this problem, we are given the mass, velocity, and tension (which is equal to the centripetal force). We need to find the radius (length of the string).

First, we rearrange the formula to solve for r:

r = (m * v^2) / F

Substituting the given values:

r = (m * v^2) / F
r = (m * 1.6^2) / 7.7

Next, we need the mass of the object. However, the problem doesn't provide the mass directly. Since the string is said to have a negligible mass and the ball is suspended from it, we can assume that the mass of the string is also negligible. Therefore, we can assume the mass of the object is equal to the tension force divided by the acceleration due to gravity (9.8 m/s^2).

m = F / g
m = 7.7 / 9.8

Finally, substitute the value of m in the equation for r:

r = ((7.7 / 9.8) * 1.6^2) / 7.7

By evaluating this expression, we can find the length of the string.