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.
95.528 is the correct ans....
can you show me the steps?thx
To estimate the rotational speed at which the motor will exhibit the largest vertical vibration, we need to understand the relationship between the sinking depth of the motor and the rotational speed.
When the motor is rotating at a high speed, it can create a centrifugal force that pushes the motor away from its center of rotation. This force can cause the motor to sink into the rubber pad, leading to vertical vibration.
The sinking depth h of the motor can be related to the angular velocity ω (in radians per second) of the motor using the following equation:
h = Aω^2
Where A is a constant that depends on the material properties of the rubber pad and the motor.
To find the angular velocity ω at which the motor exhibits the largest vertical vibration, we need to differentiate the sinking depth equation with respect to ω, and set the derivative equal to zero. This will give us the value of ω at the maximum sinking depth:
dh/dω = 2Aω = 0
Solving this equation, we find that ω = 0. This implies that the motor will exhibit the largest vertical vibration when it is not rotating at all.
Therefore, the estimated rotational speed in RPM at which the motor will exhibit the largest vertical vibration is 0 RPM.