The device shown can detect when the angular velocity and angular acceleration of a rigid...

The device shown can detect when the angular velocity and angular acceleration of a rigid body B achieve a combination of specified values. The device works using the principle that the vibration amplitude of the mass P depends on both the angular velocity and angular acceleration of the rigid body. When the angular velocity ωB and angular acceleration αB reach the appropriate combination, the mass P will contact the touch sensor, thus signaling that the specified values have been reached. With this as background, assume that the rigid body rotates in the horizontal plane with angular velocity ωB and angular acceleration αB, and that the mass P is constrained to move in the slot, which is at a distance d from the center of the disk. In addition, a linear elastic spring of constant k is attached to the mass such that the spring is undeformed when the mass is at s = 0.

(a) Derive the equation of motion for the mass P with s as the dependent variable.

(b) Assuming that the mass P is released from rest at s = 0, find the solution to the equation of motion found in (a), knowing that solution to ordinary differential equations of the type

is given by

where C1 and C2 are constants determined from the initial conditions and D is a known constant.

(c) Using the solution for s(t) found above, for given values of d, k, m, ωB, and αB, determine the maximum distance from s = 0 that the mass P achieves in one cycle.

(d) For a disk-shaped rigid body B whose diameter is 1.5 m, specify the mass of P (treat it as a particle), the spring constant k, the length h, and the distance d so that the touch sensor can detect when the rigid body reaches an angular velocity ωcrit = 100 rpm for a constant angular acceleration αB = 1 rad/s2.

Figure DP9.2