A 21.0-g metal cylinder is placed on a turntable, with its center 90 cm from the turntable's center. The coefficient of static friction between the cylinder and the turntable's surface is ìs = 0.84. A thin, massless string of length 90 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 70 rpm (rotations per minute).

Question

A 21.0-g metal cylinder is placed on a turntable, with its center 90 cm from the turntable's center. The coefficient of static friction between the cylinder and the turntable's surface is ìs = 0.84. A thin, massless string of length 90 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 70 rpm (rotations per minute).

Answer 1

tension=mass*w^2 r

where
w in rad /sec

Answer 2

Bob I have already tried that, but that does not give correct answer.A similar problem with different values have the following:

If the mass in this problem is 18 g and radius is 75 cm and mu s is 0.87 and angular velocity is 45 rpm then the tension is 0.146 N. I have no clue to get that answer, I have tried so many different ways all failed.

Answer 3

To calculate the tension in the string, we need to consider the forces acting on the cylinder.

First, let's calculate the acceleration of the cylinder using the angular velocity provided. We can convert 70 rpm to radians per second by using the conversion factor:

angular velocity (in rad/s) = angular velocity (in rpm) * (2π rad/1 min) * (1 min/60 s)

angular velocity (in rad/s) = 70 rpm * (2π rad/1 min) * (1 min/60 s)

Now we can calculate the acceleration:

acceleration = (angular velocity)^2 * radius

where the radius is given as 90 cm or 0.9 m. Substituting the values, we get:

acceleration = (70 rpm * (2π rad/1 min) * (1 min/60 s))^2 * 0.9 m

Next, let's calculate the net force acting on the cylinder. The net force is the difference between the centripetal force and the frictional force. The centripetal force is given by:

centripetal force = mass * acceleration

where the mass is given as 21.0 g or 0.021 kg. Substituting the values, we get:

centripetal force = 0.021 kg * acceleration

The frictional force can be calculated using the coefficient of static friction. The maximum frictional force is given by:

maximum static frictional force = coefficient of static friction * normal force

The normal force is equal to the weight of the cylinder, which can be calculated as:

weight = mass * gravitational acceleration

where the gravitational acceleration is approximately 9.8 m/s^2. Substituting the values, we get:

weight = 0.021 kg * 9.8 m/s^2

Finally, substituting the maximum static frictional force and the centripetal force into the net force equation, we have:

net force = centripetal force - maximum static frictional force

To find the tension in the string, we consider the equilibrium of forces. At any instant, the tension in the string provides the centripetal force required to keep the cylinder in circular motion. Therefore, the tension in the string is equal to the net force acting on the cylinder.

So, to calculate the tension in the string when the angular velocity is 70 rpm, we substitute the calculated values into the net force equation:

Tension in the string = net force = centripetal force - maximum static frictional force

Remember to convert the units to the appropriate form before plugging in the values into the equation.