MAE 202 – Mechanics of Particles and Rigid Bodies II: Dynamics
Homework 8 – RB Kinematics (Absolute Motion Analysis)
Instructions: Please be neat and organized – points may be deducted if the homework is
unorganized or unclear or if the scanned PDF is not clear. Use the suggested template for the
solutions (see the last page of the homework for the template). In every problem, clearly choose
and draw the origin and the positive directions of the axes/reference frame. Also draw the
particle(s) at a generic/generalized locations in the reference frame, if needed for the problem. It
will help you get the correct signs for various terms of the equations of motion.
Problem 1: (Use absolute motion analysis) At the instant shown,
end 𝐴 of the rod has the velocity and acceleration as shown.
Determine the angular acceleration of the rod and acceleration of
end 𝐵 of the rod.
Ans: 𝛼 = −3.67 𝑟𝑎𝑑/𝑠 2 , 𝑎𝐵 = −26.7 𝑚/𝑠 2 (F16-19)
Note: By “use absolute motion analysis”, we mean first relate the
position of end A and the position of end B (at a generic instant,
not the instant shown). Then differentiate that equation to relate
their velocities and then differentiate again to relate their
accelerations. Similarly, relate the angle of the rod to the position
of end A, and differentiate twice to relate angular acceleration of
the rod to the acceleration of end A.
𝜃
Problem 2: (Use absolute motion analysis) At the instant shown,
the disk of radius 3 𝑓𝑡 has an angle 𝜃, it is rotating with an
angular velocity 𝜔 and has an angular acceleration 𝛼. Determine
the velocity and acceleration of the cylinder 𝐵 at this instant. Your
answers can only be in terms of 𝜃, 𝜔 and 𝛼. Neglect the size of the
pulley at 𝐶.
Ans: 𝑣𝐵 = −
15𝜔 sin 𝜃
√34−30
and 𝑎𝐵 = −
cos 𝜃
15(𝛼 sin 𝜃+𝜔2 cos 𝜃)
√34−30 cos 𝜃
+
𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
3
(34−30 cos 𝜃)2
Hint: The length of the rope is a constant, its derivatives are 0. Express this length in terms of
angle 𝜃 and the position of B.
MAE 202 – Mechanics of Particles and Rigid Bodies II: Dynamics
Problem 3: (Use absolute motion analysis) The end 𝐴 of the bar 𝐴𝐵
is moving downward along the slotted guide with a constant speed 𝒗𝑨 .
The length of the bar is 𝑙 and it is sliding on a fixed circular surface of
radius 𝑟 centered at the origin. Determine the velocity of point 𝐵 and
angular velocity of bar 𝐴𝐵 as a function of the position 𝑦 and its
derivatives.
Ans: 𝑣𝐵,𝑥 =
𝑙𝑟𝑣𝐴
𝑦2
, 𝑣𝐵,𝑦 = 𝑣𝐴 (−1 +
𝑙
√𝑦 2 −𝑟 2
−
𝑙√𝑦 2 −𝑟 2
𝑦2
), 𝜔 =
𝑟𝑣𝐴
𝑦√𝑦 2 −𝑟 2
Hint: After defining origin and axes, write a relation between the
coordinates of point B (𝑥𝐵 and 𝑦𝐵 ) and the distance 𝑦. You may have to write the relation between
𝑦 and 𝜃 first. Then differentiate.
Problem 4: (Use absolute motion analysis) The
circular cam rotates about the fixed point 𝑂 with
a constant angular velocity 𝜔. Determine the
velocity the follower rod 𝐴𝐵 as a function of
time.
Ans: 𝑣𝐵 = −𝑑𝜔 sin 𝜔𝑡 −
𝑑2 𝜔 sin(2𝜔𝑡)
2√(𝑅+𝑟)2 −𝑑2 sin2 𝜔𝑡
Problem 5: (Use absolute motion analysis) As the end 𝐴
of the bar is pulled to the right with the velocity 𝑣, the bar
slides on the surface of the fixed half cylinder of radius 𝑟.
Determine the angular velocity 𝜔 = 𝜃̇ of the bar in terms
of 𝑥 and 𝑣.
Ans: 𝜔 = −
𝑣𝑟
𝑥√𝑥 2 −𝑟 2
MAE 202 – Mechanics of Particles and Rigid Bodies II: Dynamics
Problem 6: (Use absolute motion analysis) At a given instant the
center of the roller 𝐴 on the bar has the velocity and acceleration
shown. The length of the bar is 0.6 𝑚. The radius of both the rollers
is the same (and is in fact not required to solve the problem!)
Determine the magnitude of velocity and acceleration of the center
of the roller 𝐵 when the angle of the bar with the vertical, 𝜃 = 30o .
Ans: 𝑣𝐵 = 4 𝑚/𝑠 and 𝑎𝐵 = −24.8 𝑚/𝑠 2
Hint: In absolute motion analysis, we relate the positions of the
system at a generic location. Hence, do not assume that the angle 𝜃
that the bar makes with the vertical is always 30 degrees. It changes with time, and its derivative
is not zero. It is 30 degrees only at this instant. However, the angle of the plane on which roller B
moves with respect to the horizontal is always 30 degrees.
Problem 7: (Use absolute motion analysis) At the
given instant the slider block 𝐴 is moving toward the
right with the velocity and acceleration shown.
Determine the velocity and acceleration of the point B
the instant shown.
Ans: 𝑣⃗𝐵 = 4 𝑖̂ 𝑚/𝑠, 𝑎⃗𝐵 = (6 +
8
) 𝑖̂ − 8𝑗̂ 𝑚/𝑠 2
√3
Hint: In absolute motion analysis, we relate the positions
of the system at a generic location.
Do not assume that the angle 𝜃 of the bar is always 30 degrees. It changes with time, and its
derivative is not zero. Angle 𝜃 is 30 degrees only at this instant.
Similarly, B is not always at the top of the circle, you can assume that CB makes an angle 𝜙 with
respect to vertical. Angle 𝜙 is 0 only at this instant. 𝐶 is the center of the circular slot in which
the point 𝐵 moves.
After choosing an origin at a “good” location, write a relation between the generic coordinates
of point B (𝑥𝐵 and 𝑦𝐵 ) and the coordinate 𝑥𝐴 of point A. Then differentiate. You may find that
it’s easier to separately write the relationship between 𝑥𝐴 and 𝜙, and 𝑥𝐵 and 𝜙, and 𝑦𝐵 and 𝜙.
MAE 202 – Dynamics
Homework Assignment #____
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Problem #____ – Final Solution
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