Concept explainers
Two rotating rods in the vertical plane are connected by a slider block P of negligible mass. The rod attached at A has a mass of 0.8 kg and a length of 160 mm. Rod BP has a mass of 1 kg and is 200 mm long and the friction between block P and AE is negligible. The motion of the system is controlled by a couple M applied to bar BP. Knowing that at the instant shown rod BP has an angular velocity of 20 rad/s clockwise and an angular acceleration of 80 rad/s2 clockwise, determine (a) the couple M, (b) the components of the force exerted on AE by block P.
Fig. P16.141 and Fig. P16.142
(a)
Find the value of couple M.
Answer to Problem 16.142P
The value of couple M is
Explanation of Solution
Given information:
The mass of the rod AE is
The mass of the rod BP is
The length of the rod AE is
The length of the rod BP is
The angular velocity is
The angular acceleration is
Calculation:
Consider the acceleration due to gravity as
Calculate the position vector
The position of P with respect to A.
The position of P with respect to B.
The position of P with respect to E.
The angular velocity of rod BP in vector form is
The angular acceleration of rod BP in vector form is
Calculate the velocity of rod BP
Substitute
Consider the relative angular velocity of rod AE as
Calculate the velocity of point P
Substitute
Resolving i and j components as shown below.
Calculate the acceleration of rod BP
Substitute
Calculate the acceleration of point P with respect to point E
Substitute
Calculate the acceleration of point P
Substitute
Resolving i and j components as shown below.
Calculate the weight of
For rod AE.
Substitute
For rod BP.
Substitute
Calculate the mass moment of inertia
For rod AE.
Substitute
For rod BP.
Substitute
Calculate the position vector
The position of mass center G with respect to the rod AE.
The position of mass center H with respect to the rod BP.
Calculate the acceleration of point G
Substitute
Calculate the acceleration of point H
Substitute
Calculate the inertial terms of the mass center
For rod AE.
For rod BP.
Calculate the effective couples at mass center
For rod AE.
For rod BP.
Sketch the Free Body Diagram of rod AE as shown in Figure 1.
Refer to Figure 1.
Apply the Equilibrium of moment about A as shown below.
Sketch the Free Body Diagram of rod BP as shown in Figure 2.
Refer to Figure 2.
Apply the Equilibrium of moment about A as shown below.
Hence, the couple M is
(b)
Find the components of the force exerted on AE by block.
Answer to Problem 16.142P
The components of the force exerted on AE by block is
Explanation of Solution
Given information:
The mass of the rod AE is
The mass of the rod BP is
The length of the rod AE is
The length of the rod BP is
The angular velocity is
The angular acceleration is
Calculation:
Refer to part (a).
The components of the force exerted on AE by block
Therefore, the components of the force exerted on AE by block is
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