Concept explainers
The right leg of an athlete on a rowing machine can be modeled as a linkage as shown, where A represents the ankle (which is stationary), K the knee, and H the hip. At the instant when θ = 75°, the shank AK has an angular velocity of 1 rad/s and an angular acceleration of 1.5 rad/s2, both counterclockwise. Determine the velocity and acceleration of the hip H at this instant in time.
Fig. P15.123
Find the velocity and acceleration of the hip H at the instant.
Answer to Problem 15.123P
The velocity and acceleration of the hip H are
Explanation of Solution
Given information:
The ankle, knee and the hip are denoted by A, K, and H.
The ankle A is stationary.
The value of the angle is
The angular velocity of the shank AK is
The angular acceleration of the shank AK is
Calculation:
Show the modelled linkage as shown in Figure 1.
Refer to Figure 1.
Consider the distance KH and KA are denoted by
Show the relation between the distances KH and KA as follows:
Modify Equation (1) using Equation (2).
Substitute
Show the relative velocity for AK as follows:
Show the relative velocity for KH as follows:
Consider the position of the point K with respect to A is denoted by
Consider the position of the point H with respect to K is denoted by
Substitute
Equate the j component of the Equation (4).
Substitute
Equate i component of the Equation (4).
Substitute
Thus, the velocity at H is
Consider AK.
Show the relation between the acceleration of A and K as follows:
Substitute
Consider HK.
Show the relation between the acceleration of H and K as follows:
Substitute
Equate j component of the Equation (7).
Substitute
Equate i component of the Equation (7).
Substitute
Thus, the acceleration at H is
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