A 1.62 kg mass attached to a light string

rotates on a horizontal, frictionless table. The
radius of the circle is 0.986 m, and the string
can support a mass of 32.5 kg before breaking.
The acceleration of gravity is 9.8 m/s
2
.
What maximum speed can the mass have
before the string breaks?
Answer in units of m/s

To find the maximum speed the mass can have before the string breaks, we need to consider the tension in the string. At maximum speed, the centripetal force provided by the tension in the string should be equal to the centrifugal force.

The centripetal force is given by the equation:

Fc = m * ac

Where:
Fc = Centripetal force
m = Mass of the rotating object
ac = Centripetal acceleration

The centrifugal force is given by the equation:

Fcf = m * g

Where:
Fcf = Centrifugal force
g = Acceleration due to gravity

Since the tension in the string provides the centripetal force, we can set Fc equal to Fcf:

m * ac = m * g

Rearranging the equation for ac, we get:

ac = g

Now, the centripetal acceleration can be calculated using the formula:

ac = v^2 / r

Where:
v = Velocity of the object
r = Radius of the circle

Substituting ac = g into the equation, we get:

g = v^2 / r

Rearranging the equation for v, we find:

v = sqrt(g * r)

Now we can substitute the given values to calculate the maximum speed:

g = 9.8 m/s^2
r = 0.986 m

v = sqrt(9.8 * 0.986)
v ≈ 3.14 m/s

Therefore, the maximum speed the mass can have before the string breaks is approximately 3.14 m/s.