To simplify this process, consider the following diagram (Figure 2). The location of the piston is...

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To simplify this process, consider the following diagram (Figure 2). The location of the piston is represented by the point P. Suppose the crank has a circular rotation radius of 23 cm (distance from the origin to point A). The connecting rod (line AP) has a length of 40 cm. The rotation occurs counterclockwise, as (theta) increases through positive values at a rate of 360 revolutions per minute. The angle between line AP and the x-axis is represented by (alpha). a) Find the angular velocity of the connecting rod, that is find , in radians per second, when radians. Hint: Use the law of sines to help create a formula involving the two angles and two known sides, then differentiate implicitly. Also note is given in revolutions per minute, this needs to be converted to radians per second. It is important to calculate the forces exerted on the piston as the crank rotates. The forces produced are determined by the linear speed at which P travels along the x-axis. In order to calculate this speed, the distance must first be determined, and then can be computed. b) Express the distance as a function of. In other words, find the length of the line segment OP (from the origin to point P) as a function of for r = 23 cm and |AP| = 40cm. Hint: Use the law of cosines to relate x and theta, the equation will be quadratic which you must solve for x. c) Use your result from part b, find an expression for the velocity of the pin P d) What is the maximum speed of the piston? What are the units for? Intuitively, at what point (in the motion along the x-axis) does this maximum speed occur? To answer this question, only use a graphical approach, instead of maximizing through finding critical points. Graph your equation from part c in desmos, find an appropriate viewing window to "see" the graph, click on the maximum value so desmos labels the point. Print/screen-shot the graph and include in your submission. To simplify this process, consider the following diagram (Figure 2). The location of the piston is represented by the point P. Suppose the crank has a circular rotation radius of 23 cm (distance from the origin to point A). The connecting rod (line AP) has a length of 40 cm. The rotation occurs counterclockwise, as (theta) increases through positive values at a rate of 360 revolutions per minute. The angle between line AP and the x-axis is represented by (alpha). a) Find the angular velocity of the connecting rod, that is find , in radians per second, when radians. Hint: Use the law of sines to help create a formula involving the two angles and two known sides, then differentiate implicitly. Also note is given in revolutions per minute, this needs to be converted to radians per second. It is important to calculate the forces exerted on the piston as the crank rotates. The forces produced are determined by the linear speed at which P travels along the x-axis. In order to calculate this speed, the distance must first be determined, and then can be computed. b) Express the distance as a function of. In other words, find the length of the line segment OP (from the origin to point P) as a function of for r = 23 cm and |AP| = 40cm. Hint: Use the law of cosines to relate x and theta, the equation will be quadratic which you must solve for x. c) Use your result from part b, find an expression for the velocity of the pin P d) What is the maximum speed of the piston? What are the units for? Intuitively, at what point (in the motion along the x-axis) does this maximum speed occur? To answer this question, only use a graphical approach, instead of maximizing through finding critical points. Graph your equation from part c in desmos, find an appropriate viewing window to "see" the graph, click on the maximum value so desmos labels the point. Print/screen-shot the graph and include in your submission.