課程名稱︰動力學 課程性質︰必修 課程教師︰陽毅平 開課學院：工學院 開課系所︰機械系 考試日期（年月日）︰2013.5.14 考試時限（分鐘）：3小時 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 圖片連結：http://ppt.cc/WxRC 1. A triangular body m2 of mass M (kg) rests on a frictionless, horizontal surface. At t=0, a mass m1 of m (kg) is released from rest as shown. It's located just slightly to the left of center so that it can slide down the left surface. Assume zero friction between m1 and m2. (a) Define inertial coordinates and body fixed coordinates on the triangular body.(2%) (b) Write equations of motion for m1. (5%) (c) Write equations of motion for m2. (5%) (d) If M=2m， L = 11/ 12√(3) (meter)， what is the velocity vector of the mass m1 when it hits the ground? (express the velocity as a function of g) (8%) 2. A particle is placed just to the right of the top of a smooth semicircular hill. Because it's infinitesimally to the right and the surface is frictionless， it will start to slide down the right slope. (a) Define inertial coordinates and body-fixed coordinates on the particle. (5%) (b) Write equations of motion for the particle. (5%) (c) Determine the particle speed when it's at an arbitrary value of θ before it's losing contact with the surface.(5%) (d) Determine the angle where the particle just loses contact with the surface. (5%) 3. A ball with mass m is attached to a rigid massless rod of length L that can freely pivot about its end O in the horizontal plane. The ball initially rotates counterclockwise about 0 with an angular speed of ω0， and it moves over a rough surface with a coefficient of friction μ (a) Define inertial coordinates and body-fixed coordinates. (2%) (b) Write the position and velocity vectors of the ball. (5%) (c) Draw a free-body diagram to describe all the forces applied on the ball， and write the force vector. (5%) (d) Use the impulse-momentum principle to determine the time for the. ball to come to a stop. (8%) 4. A bar of length L and negligible mass connects a particle of mass m and the center of a cart of mass M. The cart is initially at rest while the bar is released from rest at θ=π/2. The mass of wheels are neglected， and all joints and surfaces are smooth. (a) Derive the equations of motion of the cart. (5%) (b) Derive the equations of motion of the particle. (5%) 5. A small bead of mass m is carried by a circular hoop of radius r which rotates about a fixed vertical axis. The friction is neglected， but it is assumed that a small amount of friction is present to damp out any motion of the bead relative to the hoop once a constant angular speed has been established. → → → The coordinate system (a ，a ，a ) is fixed on the hoop， while 　　　　　　　　　　　　 1 2 3 　　　　　　　　　　　 → → → the coordinate system (b ，b ，b ) is fixed on the bead and the angle θ 　　　　　　　　　　　　1 2 3 　　　　　　　　　　　　　　　　　　　　　 → → → is measured "relative" to the coordinate (a ，a ，a ). 　　　　　　　　　　　　　　　　　　　　　　1 2 3 　　　　　　　　　　　　　　　　　　　　 　 → → → (a) Write the angular velocities of the coordinate system (a ，a ，a ) 　　　　　　　　　　　　　　　　　　　　 　　1 2 3 　　 → → → and (b ，b ，b ). (5%) 1 2 3 (b) Write the equations of motion of the bead. (10%) 6. A m-kg collar C rides along a horizontally arm AB. A spring k connects C to A. The arm AB rotates about a fixed point A. The coefficient of dynamic friction isμk﹒ (a) Write the equations of motion of the collar. (10%) (b) At the illustrated instant， the angular velocity is constant at . √(3)g 2 g ‥ g ω (rad/sec)， r ω = ─── ， r ω = ─ ，and r = ─ 0 0 2　　　　　 0 4 2 where g is the gravitional acceleration， what is "the total force" exerted by the spring constant on the collar? (5%) --

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