A bug is 12 cm from the center of a turntable that is rotating with a frequency of 45 rev/min. What minimum coefficient of friction is required so that the bug stays on the turntable?
R = 0.12 m
omega = 45 rev/min *1 min/60s * 2 pi radians/rev= 4.71 rad/s
Ac = R omega^2 = 2.66 m/s^2
m Ac = mu m g
so
mu = Ac/g = 2.66/9.81
A machine of a 100N from a distance 10N is use to move another force 500N to distance 10M.calculate the efficiency of machine
To find the minimum coefficient of friction required for the bug to stay on the turntable, we need to consider the centripetal force acting on the bug.
The centripetal force is provided by the friction force between the bug and the turntable. The equation for centripetal force is:
F = m * ω^2 * r
Where:
- F is the centripetal force
- m is the mass of the bug
- ω is the angular velocity (in radians per second)
- r is the distance of the bug from the center of rotation
First, let's convert the given frequency from rev/min to radians per second:
ω = 2π * f
Where:
- ω is the angular velocity (in radians per second)
- f is the frequency (in revolutions per minute)
Substituting the values:
ω = 2π * 45 rev/min
= 2π * (45/60) rev/s
= 2π * (0.75) rev/s
= 1.5π rev/s
Next, let's convert the distance from cm to meters:
r = 12 cm
= 0.12 m
Now, we can substitute these values into the centripetal force equation and solve for the minimum coefficient of friction:
F = m * (1.5π)^2 * 0.12
To find the minimum coefficient of friction, we need to set the frictional force equal to the maximum static friction force, which is given by:
F_friction = µ * N
Where:
- F_friction is the frictional force (equal to the centripetal force)
- µ is the coefficient of friction
- N is the normal force (equal to the weight of the bug)
The normal force is equal to the weight of the bug, which can be calculated as:
N = m * g
Where:
- N is the normal force
- m is the mass of the bug
- g is the acceleration due to gravity (9.8 m/s^2)
Substituting for N:
F_friction = µ * (m * g)
Setting the centripetal force equal to the frictional force:
m * (1.5π)^2 * 0.12 = µ * (m * g)
Simplifying the equation by canceling out the mass:
(1.5π)^2 * 0.12 = µ * g
Finally, solve for µ:
µ = ((1.5π)^2 * 0.12) / g
Substituting the value of g (9.8 m/s^2) and simplifying:
µ ≈ ((1.5π)^2 * 0.12) / 9.8
≈ 0.085
Therefore, the minimum coefficient of friction required for the bug to stay on the turntable is approximately 0.085.