Concept explainers
Knowing that at the instant shown the rod attached at A has an angular velocity of 5 rad/s counterclockwise and an angular acceleration of 2 rad/s2 clockwise, determine the angular velocity and the angular acceleration of the rod attached at B.
Fig. P15.176
The angular velocity and the angular acceleration of the rod at B.
Answer to Problem 15.176P
The angular velocity of the rod at B is
The angular acceleration of the rod at B is
Explanation of Solution
Given information:
The angular velocity of the rod at A is
The angular acceleration of the rod at A is
Calculation:
Consider the triangle ABP to get the values of AP and BP.
Calculate the angle ABP
Consider the sum of the angles is
Calculate the angle APB
Substitute
Apply law of sine for the triangle ABP as shown below.
Substitute
Calculate the position vectors
Position of P with respect to A.
Position of P with respect to B.
Provide the angular velocity of each link in vector form as shown below.
Provide the angular acceleration of each link in vector form as shown below.
Calculate the velocity of point P
Here,
Calculate the velocity the point P on the rod AP
Substitute
Calculate the acceleration at the point P
Substitute
Calculate the relative velocity component
Here, u is the velocity of the point P relative to BP.
The angle of relative velocity with respect to the rotating rod BP is
Resolve the relative velocity along x and y directions.
Calculate the relative acceleration component
Here,
Resolve the relative velocity along x and y directions.
Calculate the velocity component
Substitute
Calculate the angular velocity
Substitute
Resolving i and j components as shown below.
For i component.
For i component.
Solving Equations (3) and (4) to get the values as shown below.
Hence, the angular velocity of the rod at B is
Calculate the relative velocity
Substitute
Calculate the acceleration of the point P.
Calculate the acceleration component
Substitute
Calculate the Coriolis component of acceleration
Substitute
Calculate the angular acceleration
Substitute
Resolving i and j components as shown below.
For i component.
For j component.
Solving Equations (6) and (7) to get the values as shown below.
Hence, the angular acceleration of the rod at B is
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