A 21.0-g metal cylinder is placed on a turntable, with its center 85 cm from the turntable's center. The coefficient of static friction between the cylinder and the turntable's surface is ìs = 0.83. A thin, massless string of length 85 cm connects the center of the turntable to the cylinder, and initially, the string has zero tension in it. Starting from rest, the turntable very slowly attains higher and higher angular velocities, but the turntable and the cylinder can be considered to have uniform circular motion at any instant. Calculate the tension in the string when the angular velocity of the turntable is 80 rpm (rotations per minute).

To calculate the tension in the string when the angular velocity of the turntable is 80 rpm, we need to consider the forces acting on the metal cylinder.

1. The gravitational force acting on the cylinder: The weight of the cylinder can be calculated using the equation F = mg, where m is the mass of the cylinder and g is the acceleration due to gravity. Given that the mass of the cylinder is 21.0 g, we convert it to kilograms (1 kg = 1000 g) and multiply it by the acceleration due to gravity (approximately 9.8 m/s^2) to find the weight.

2. The centripetal force acting on the cylinder: When the cylinder is rotating in circular motion, it experiences a centripetal force directed towards the center of the circle. This force can be calculated using the equation F = mω^2r, where m is the mass of the cylinder, ω is the angular velocity (in radians per second), and r is the distance between the center of the turntable and the center of the cylinder.

However, before calculating the centripetal force, we need to determine the static friction force between the cylinder and the turntable's surface. The static friction force prevents the cylinder from slipping on the turntable as it rotates.

The formula for static friction is Fs ≤ μsN, where Fs is the static friction force, μs is the coefficient of static friction, and N is the normal force.

3. The normal force acting on the cylinder: The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is equal to the weight of the cylinder acting vertically downwards.

Now we can calculate the tension in the string.

First, find the weight of the cylinder:
weight = mass * acceleration due to gravity
weight = 21.0 g * (1 kg / 1000 g) * 9.8 m/s^2

Next, calculate the centripetal force:
centripetal force = mass * angular velocity^2 * distance
centripetal force = 21.0 g * (1 kg / 1000 g) * (80 rpm * (2π rad/min) * (1 min / 60 s))^2 * 0.85 m

Then, determine the normal force (equal to the weight of the cylinder):
normal force = weight

Finally, calculate the static friction force and the tension in the string:
static friction force = coefficient of static friction * normal force
tension in the string = centripetal force + static friction force

Substitute the values into the equations to find the tension in the string.