View this principle with an animation on YouTube and an animated Desmos graph. To simplify this...

Transcribed Image Text:

View this principle with an animation on YouTube and an animated Desmos graph. 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 a (alpha). da dt A P(x,0) Figure 2 a) Find the angular velocity of the connecting rod, that is find at, in radians per second, when radians. Hint: Use the law of sines to help create a formula involving the = de is given in two angles and two known sides, then differentiate implicitly. Also note dt 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 can be computed. calculate this speed, the distance x must first be determined, and then d dt b) Express the distance x as a function of 0. In other words, find the length of the line segment OP (from the origin to point P) as a function of 0 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. dx c) Use your result from part b, find an expression for the velocity at of the pin P dt View this principle with an animation on YouTube and an animated Desmos graph. 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 a (alpha). da dt A P(x,0) Figure 2 a) Find the angular velocity of the connecting rod, that is find at, in radians per second, when radians. Hint: Use the law of sines to help create a formula involving the = de is given in two angles and two known sides, then differentiate implicitly. Also note dt 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 can be computed. calculate this speed, the distance x must first be determined, and then d dt b) Express the distance x as a function of 0. In other words, find the length of the line segment OP (from the origin to point P) as a function of 0 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. dx c) Use your result from part b, find an expression for the velocity at of the pin P dt