Discussion and hints:

Let’s first take a look at the motion of the mechanism, as shown in the simulation results below.

The kinematics for this problem are a little complicated. Recall that we studied these types of “constrained motion” kinematics problems back in Section 1.D of the lecture book. Here, we will now get to use the skills developed there in solving this problem in kinetics.

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw individual free body diagrams for bodies A and B.

Step 2: Kinetics (Newton’s 2nd Law)
Write down the appropriate Newton’s 2nd law equations for blocks A and B.

Step 3: Kinematics
Here, we will use the assumption that the cable length, L, does not change as the system moves (the cable does not stretch, go slack or break). For this, write L in terms of the motion variables s_A and s_B. You might want to review the material and examples from Section 1.D of the lecture book in doing this. Differentiate this relation once to relate the speeds of A and B, and then, once again, to relate the accelerations of A and B. NOTE: Be sure to use the same sign conventions for forces as for accelerations. For example, if you choose positive x as being to the right, then a_A = – s_A_ddot.

Step 4: Solve
At this point you will have three equations involving the tension in the cable, the acceleration of A and the acceleration of B. Solve these three equations for those three unknowns.

Any questions??

Discussion and hints:

Since all motion is in a horizontal plane and with the contact surface of the slot with P being smooth, the speed of P will remain constant throughout. And, with the speed being a constant, there will be only a normal (centripetal) component of acceleration for P. (If you do not see this from the beginning, no worries. Your analysis will show you this.) This is seen in the animation provided below.

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw a free body diagram for particle P for the position of interest: x = 0. Please be reminded that the motion of the system is in a horizontal plane. With this and with the slot being smooth, only the normal contact force from the slot acts on P.

Step 2: Kinetics (Newton’s 2nd Law)
Write down the t – and n -components of the Newton’s 2nd law equations for P.

Step 3: Kinematics
Here, we will consider the constrained motion of particle P. Use the chain rule to determine y_dot in terms of x_dot, as well as y_ddot in terms of x_ddot. For the instant of interest, i = e_t and j = e_n.

Step 4: Solve
From your kinetics and kinematics equations, solve for the normal contact force, N.

Any questions??

Ask and answer questions here. You learn both ways.

DISCUSSION and HINTS

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw single free body diagram (FBD) for the entire system.

Step 2: Kinetics (Work/energy equation)
Consider all of the external forces that you included in your FBD above. Which forces, if any, do non-conservative work on that system? If there are no such forces, then energy is conserved. Write down the expressions for kinetic and potential energy for the initial and final states of the motion.

Step 3: Kinematics
At this step, you need to relate the speeds of blocks A and B at the second state. Consider using the instant center (IC) approach for the rigid link for position 2. Block B moves along a horizontal surface, and Block A rides on a vertical guide. Where do the perpendiculars to these two velocity vectors intersect? Take a look at the freeze-frame of the animation of the motion at state 2 shown below. Does that image agree with your IC analysis? And, what does this say about the speed of block B at that state?

Step 4: Solve
Combine your kinetics equation from Step 2 with your kinematics that you found in Step 3, and solve for the speed of A.

Discussion and hints:

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBD
Draw a free body diagram of the system made up of P. Which, if any, forces do non-conservative work on this system? Can you justify this from the FBD?

Step 2: Kinetics (work/energy equation)
Write down the work energy equation for P. Recall that the potential energy in a spring is 0.5*k*Δ², where Δ is the stretch/compression in the spring. Δ is NOT equal to the length of the spring. Recall that the spring is unstretched at position 2.

Step 3: Kinematics
What kinematics do you need here?

Step 4: Solve
Solve your work/energy equation for the speed of P at position 2.

Any questions??