A freely running motor rests on a thick rubber pad to reduce vibration. The motor sinks h=10 cm into the pad. Estimate the rotational speed in RPM (revolutions per minute) at which the motor will exhibit the largest vertical vibration.
I pad can be modeled as a spring
95.528
that's not the answer
To estimate the rotational speed at which the motor will exhibit the largest vertical vibration, we need to analyze the forces acting on the motor.
When the motor rotates, it experiences a centrifugal force due to its rotational motion. This force causes the motor to sink into the rubber pad, resulting in vertical vibration. The magnitude of the centrifugal force is given by:
F_cent = m * ω^2 * r
Where:
F_cent is the centrifugal force
m is the mass of the motor
ω is the angular velocity (in radians per second)
r is the distance of the motor from the axis of rotation
In this case, the sinking of the motor indicates that the centrifugal force is balanced by the normal force exerted by the rubber pad. When the motor sinks by 10 cm, it means that the normal force and the centrifugal force are equal in magnitude but opposite in direction.
We can set up an equation by equating the forces in the vertical direction:
m * g - m * ω^2 * r = 0
Where:
m is the mass of the motor
g is the acceleration due to gravity (9.8 m/s^2)
ω is the angular velocity (in radians per second)
r is the distance of the motor from the axis of rotation
Rearranging the equation, we get:
ω^2 = g / r
To estimate the rotational speed at which the motor will exhibit the largest vertical vibration, we need to determine the value of ω that maximizes the magnitude of ω. Since ω^2 is proportional to g, the largest ω will occur when g is maximized.
At the surface of the Earth, g is constant and its maximum value is 9.8 m/s^2. Therefore, to maximize the magnitude of ω, we need to minimize the distance r.
Since the motor sinks 10 cm into the rubber pad, the distance r is equal to the depth the motor sinks. Thus, r = 0.1 m.
Now we can substitute the values into the equation and calculate ω:
ω^2 = (9.8 m/s^2) / (0.1 m)
ω^2 = 98 rad^2/s^2
ω ≈ 9.90 rad/s
To convert ω to RPM (revolutions per minute), we can use the conversion factor:
1 rad/s ≈ 9.5493 RPM
Therefore, the rotational speed at which the motor will exhibit the largest vertical vibration is approximately:
ω ≈ 9.90 rad/s * 9.5493 RPM/rad ≈ 94.42 RPM
So, the estimated rotational speed is approximately 94.42 RPM.