A 25.0-g metal cylinder is placed on a turntable, with its center 70 cm from the turntable's center. The coefficient of static friction between the cylinder and the turntable's surface is μs = 0.71. A thin, massless string of length 70 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
To calculate the tension in the string when the angular velocity of the turntable is 80 rpm, we can use the concept of centripetal force and equate it to the tension in the string.
Step 1: Convert the angular velocity from revolutions per minute (rpm) to radians per second (rad/s).
Given:
Angular velocity, ω = 80 rpm
We know that:
1 revolution = 2π radians
1 minute = 60 seconds
So, to convert from rpm to rad/s:
80 rpm * (2π radians/1 revolution) * (1 minute/60 seconds) = ω rad/s
Step 2: Calculate the acceleration due to uniform circular motion.
The acceleration due to uniform circular motion can be calculated using the formula:
a = r * ω^2
Given:
Radius, r = 70 cm = 0.7 m
Angular velocity, ω (in rad/s) = calculated in Step 1
Substituting the values:
a = 0.7 m * (ω rad/s)^2
Step 3: Calculate the net force acting on the cylinder.
The net force acting on the cylinder is the product of its mass and acceleration:
F_net = m * a
Given:
Mass of the cylinder, m = 25.0 g = 0.025 kg
Substituting the values:
F_net = 0.025 kg * (0.7 m * (ω rad/s)^2)
Step 4: Calculate the maximum static friction force between the cylinder and the turntable's surface.
The maximum static friction force, F_max, can be calculated using the equation:
F_max = μs * N
Given:
Coefficient of static friction, μs = 0.71
Normal force, N = mg
Using the acceleration due to gravity, g = 9.8 m/s^2:
N = 0.025 kg * 9.8 m/s^2
Substituting the values:
F_max = 0.71 * (0.025 kg * 9.8 m/s^2)
Step 5: Determine the tension in the string.
At maximum static friction, the tension in the string, T = F_max.
Substituting the value of F_max from Step 4:
T = 0.71 * (0.025 kg * 9.8 m/s^2)
Finally, calculate the tension T to find the answer.
To calculate the tension in the string when the angular velocity of the turntable is 80 rpm, we need to first determine the acceleration of the cylinder.
The acceleration of an object in uniform circular motion is given by the equation:
a = r * ω^2
where a is the acceleration, r is the radius, and ω is the angular velocity.
In this case, the radius of the turntable is given as 70 cm. To convert this to meters, we divide by 100:
r = 70 cm / 100 = 0.7 meters
The angular velocity is given as 80 rpm. To convert this to radians per second, we multiply by 2π/60:
ω = (80 rpm) * (2π/60)
= 16π/3 radians per second
Now we can calculate the acceleration:
a = (0.7 meters) * (16π/3 radians per second)^2
≈ 74.50 meters per second squared
Next, we can calculate the force of static friction acting on the cylinder using the equation:
f_friction = μs * m * a
where μs is the coefficient of static friction, m is the mass of the cylinder, and a is the acceleration.
The mass of the cylinder is given as 25.0 g. To convert this to kilograms, we divide by 1000:
m = 25.0 g / 1000 = 0.025 kg
Now we can calculate the force of static friction:
f_friction = (0.71) * (0.025 kg) * (74.50 meters per second squared)
≈ 0.1315 Newtons
Finally, the tension in the string is equal to the force of static friction:
Tension = f_friction ≈ 0.1315 Newtons
Therefore, when the angular velocity of the turntable is 80 rpm, the tension in the string is approximately 0.1315 Newtons.