Question 15.13: The bent rod OAB rotates about the vertical OB. At the insta......
The bent rod OAB rotates about the vertical OB. At the instant considered, its angular velocity and angular acceleration are, respectively, 20 rad/s and 200 rad/s², both clockwise when viewed from the positive Y axis. The collar D moves along the rod, and at the instant considered, OD = 8 in. The velocity and acceleration of the collar relative to the rod are, respectively, 50 in./s and 600 in./s², both upward. Determine (a) the velocity of the collar, (b) the acceleration of the collar.
Frames of Reference. The frame OXYZ is fixed. We attach the rotating frame Oxyz to the bent rod. Its angular velocity and angular acceleration relative to OXYZ are therefore Ω = (-20 rad/s)j and \dot{\Omega } = (-200 rad/s²)j, respectively. The position vector of D is
r = (8 in.)(sin 30°i + cos 30°j) = (4 in.)i + (6.93 in.)j
a. Velocity \text v_D. Denoting by D^{\prime} the point of the rod which coincides with D and by F the rotating frame Oxyz, we write from Eq. (15.46)
\text v _P= \text v _{P^{\prime}}+ \text v _{P / F} (15.46)
\text v _D= \text v _{D^{\prime}}+ \text v _{D / F} (1)
where
Substituting the values obtained for \mathrm{v}_{D^{\prime}}\text{ and } \mathrm{v}_{D / F} into (1), we find
\mathrm {v}_{D}=(25 ~\mathrm{in} . / \mathrm{s}){i}+(43.3~ \mathrm{in} . / \mathrm{s}){j}+(80~ \mathrm{in} . / \mathrm{s}){k}b. Acceleration a_D. From Eq. (15.48)
a _P= a _{P^{\prime}}+ a _{p / F }+ a _c (15.48)
we write
a _D= a _{D^{\prime}}+ a _{D / F }+ a _c (2)
where
Substituting the values obtained for {a}_{D^{\prime}}, {a}_{D / {F}}, \text{ and }{a}_{c} into (2),
{a}_{D}=-\left(1300~ \mathrm{in} . / \mathrm{s}^{2}\right){i}+\left(520~ \mathrm{in} . / \mathrm{s}^{2}\right) {j}+\left(1800~ \mathrm{in} . / \mathrm{s}^{2}\right){k}