課程名稱︰動力學
課程性質︰必修
課程教師︰陽毅平 教授
開課學院：工學院
開課系所︰機械工程學系
考試日期（年月日）︰107/06/04
考試時限（分鐘）：
試題 :
圖：https://imgur.com/sH5udaL
1. A uniform bar of length L and mass m is suspended horizontally at its ends A and B by inextensible strings of equal length b. When one of the strings is cut, what is the resulting angular acceleration of the bar and the tension of string. (10%)
2. The two blades of radius R rotate counterclockwise with a constant angular velocity ω about the shaft at O mounted in the sliding block. The block is moving to the right with an acceleration of a0.
(a). Determine the acceleration of the tip A of the blade when θ = 90° in the terms of inertial coordinates. (5%)
(b). If the mass moment of inertia of each blade is (1/3)mR^2 withe respect to its end, what is the mechanical energy of the system. (5%)
(c). Is the mechanical energy conserved ? Why? (5%)
3. A metal hoop with a radius r and mass m is projected along the incline of angle θ with an initial speed v0 and angular velocity ω0, where rω0 > v0. The coefficient of kinetic friction is μk, and the mass moment of inertia about its center is mr^2.
(a). Define the inertial frame and a body-fixed coordinate system on the hoop. (2%)
(b). Write the displacement, velocity and the acceleration vectors of the center of mass. (3%)
(c). Plot the free body diagram with all the forces acting on the hoop before the hoop rolls without slip. (2%)
(d). Write equations of motion of the hoop before it rolls without slip. (4%)
(e). How long will it take for the hoop to rotate without slip? (9%)
4. A bowl with a circular cross section of radius R and mass M lies on a frictionless horizontal surface as shown in the figure. Initially the system is stationary. A small block of mass m starts at point A (θ=0) and slides down the inside of the frictionless bowl.
(a). Define the inertial frame and a body-fixed coordinate system on the small block m. (4%)
(b). Find the equations of motion of the small mass along the axes of inertial frame. (6%)
(c). Find the equations of motion of the bowl along the axes of inertial frame. (6%)
(d). What are unknowns in your equations of motion obtained in (b). and (c). . (4%)
(e). What is the velocity of the bowl as the small mass reaches the bottom of the bowl? (10%)
5. A uniform slender bar of mass m is released frome rest in thehorizontal position shown, where x = L /(2*(3^(1/2)). The coefficient of friction between the bar and edge is μ.
(a). Define the inertial frame and a body-fixed coordinate system on the bar. (4%)
(b). Write the general expressions of position, velocity, and acceleration of the center of mass before the bar slips. (5%)
(c). Write the general expressions of position, velocity, and acceleration of the center of mass after the bar slips. (3%)
(d). What is the initial angular acceleration of the bar just when it is released? (extra 7%)
(e). Write the equations of motion of the bar before it slips. (extra 6%)
(f). Determine the angle θ at which the cylinder starts to slips. (extra 12%)
6. A uniform slender bar of length L and mass m is sliding to the left on the horizontal surface with velocity v0 when it strikes the small step at O. Assume negligible rebound at the step.
(a). What is the angular velocity of the bar immediately after impact? (5%)
(b). What is the percentage energy loss after impact? (5%)
(c). Compute the minimum value of v0 which will permit the bar to pivot freely about O and just reach the standing position A with no velocity. (5%)
解答：
第一大題：https://imgur.com/UNly6Tc
第二大題：https://imgur.com/sxZHHQm
第三大題：https://imgur.com/atgQTVt
第四大題：https://imgur.com/2wD6LNj
第五大題(1)：https://imgur.com/0xXqUGL
第五大題(2)：https://imgur.com/uyhfHQE
第六大題：https://imgur.com/rHrVGs1