Concept explainers
Pin P is attached to the wheel shown and slides in a slot cut in bar BD. The wheel rolls to the right without slipping with a constant angular velocity of 20 rad/s. Knowing that x = 480 mm when θ = 0, determine the angular velocity of the bar and the relative velocity of pin P with respect to the rod when (a) θ = 0, (b) θ = 90°.
Fig. P15.154
(a)
Find the angular velocity of the bar when
Find the relative velocity of the pin P with respect to the rod when
Answer to Problem 15.154P
The angular velocity of the bar when
The relative velocity of the pin P with respect to the rod when
Explanation of Solution
Given information:
The constant angular velocity of the wheel is
The distance BA is
The distance AP is
The radius of the wheel is
Consider the relative velocity of the pin P with respect to the rod is denoted by u.
Calculation:
Show the wheel and the rod arrangement as shown in Figure 1.
Refer Figure 1.
Consider the coordinates of the point A, B, C, and P are
Calculate the value of the distance
Consider the angular velocity of the wheel AC is
Consider the angular velocity of the rod BD is
Consider the velocity of the point P with respect to point A is denoted by
Show the velocity at point P
Consider the velocity of the point P with respect to point F is denoted by
Show the velocity at point P
Equate Equation (1) and (2).
Equate the horizontal component of the Equation (3) as follows:
Take the direction towards right as positive.
Equate the vertical component of the Equation (3) as follows:
Take the direction towards downward as positive.
Calculate the value of the angle
Substitute
Modify Equation (4) as follows:
Substitute
Modify Equation (5) as follows:
Substitute
Substitute
Thus, the angular velocity of the bar when
Calculate the relative velocity of the pin P with respect to the rod when
Substitute
Thus, the relative velocity of the pin P with respect to the rod when
(b)
Find the angular velocity of the bar when
Find the relative velocity of the pin P with respect to the rod when
Answer to Problem 15.154P
The angular velocity of the bar when
The relative velocity of the pin P with respect to the rod when
Explanation of Solution
Given information:
Calculation:
Consider the value of the angle
Refer to Part (a).
Calculate the value of the angle
Substitute
Modify Equation (4) as follows:
Substitute
Thus, the relative velocity of the pin P with respect to the rod when
Modify Equation (5) as follows:
Substitute
Thus, the angular velocity of the bar when
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