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).
I converted grams to Kg, cm to m and rpm to radians..etc after that my calculations give me wrong answer, can someone help me.Maybe the coefficient of static friction is not to be used in calculations because metal cylinder and turntable are both in motion........aghhh lost ;(
To calculate the tension in the string when the angular velocity of the turntable is 70 rpm, we can follow these steps:
Step 1: Convert the angular velocity from rpm to radians per second.
To convert from rpm to radians per second, multiply the angular velocity by (2π / 60) since there are 2π radians in a full revolution and 60 seconds in a minute.
Angular velocity = 70 rpm * (2π / 60) = (7π / 3) rad/s
Step 2: Calculate the centripetal acceleration of the cylinder.
The centripetal acceleration can be calculated using the formula a = rω², where r is the distance of the cylinder from the turntable's center and ω is the angular velocity.
Distance, r = 90 cm = 0.9 m
Centripetal acceleration, a = (0.9 m) * ((7π / 3) rad/s)² = (63π² / 30) m/s²
Step 3: Calculate the net force acting on the cylinder.
The net force acting on the cylinder is equal to the product of its mass and the centripetal acceleration.
Mass, m = 21.0 g / 1000 = 0.021 kg
Net force = (0.021 kg) * (63π² / 30) m/s² = (3.35π² / 10) N
Step 4: Calculate the frictional force acting on the cylinder.
The frictional force can be calculated using the formula F_friction = μ_s * N, where μ_s is the coefficient of static friction and N is the normal force. In this case, the normal force is equal to the weight of the cylinder since it is not accelerating vertically.
Weight, W = mass * acceleration due to gravity = (0.021 kg) * (9.8 m/s²) = 0.2058 N
Frictional force = (0.84) * (0.2058 N) = 0.1725 N
Step 5: Calculate the tension in the string.
Since the turntable is slowly attaining higher and higher angular velocities, the tension in the string becomes equal to the frictional force acting on the cylinder.
Tension in the string = 0.1725 N
Therefore, the tension in the string when the angular velocity of the turntable is 70 rpm is approximately 0.1725 N.
To find the tension in the string when the angular velocity of the turntable is 70 rpm, we need to consider the forces acting on the metal cylinder.
At any instant, the tension in the string provides the centripetal force required for the circular motion of the cylinder. The force of static friction between the cylinder and the turntable's surface also provides a centripetal force.
To start solving the problem, let's first determine the acceleration of the cylinder. The centripetal acceleration of an object in circular motion is given by:
ac = r * ω^2
where ac is the acceleration, r is the radius, and ω is the angular velocity. Convert the angular velocity from rpm to radians/second:
ω = (70 rpm) * (2π rad/min) * (1/60 min/s) = (70 * 2π / 60) rad/s
Now, convert the radius from centimeters to meters:
r = 90 cm = 0.9 m
The acceleration of the cylinder is:
ac = (0.9 m) * [(70 * 2π / 60) rad/s]^2
Next, we can calculate the force of static friction. The maximum force of static friction can be found using:
f_s_max = μ_s * m * g
where μ_s is the coefficient of static friction, m is the mass of the cylinder, and g is the acceleration due to gravity. Convert the mass to kilograms:
m = 21.0 g = 0.021 kg
We have already converted grams to kilograms, so we can directly use the value of μ_s = 0.84. The force of static friction is:
f_s_max = (0.84) * (0.021 kg) * 9.8 m/s^2
Now, let's equate the two centripetal forces acting on the cylinder:
f_s_max = m * ac
(0.84) * (0.021 kg) * 9.8 m/s^2 = (0.021 kg) * [(0.9 m) * [(70 * 2π / 60) rad/s]^2]
Solve for the tension in the string, which is equal to the force of static friction:
f_s_max = (0.021 kg) * [(0.9 m) * [(70 * 2π / 60) rad/s]^2]
Finally, substitute the known values into the equation and solve for the tension in the string.