Mechanics Exercise Class Ⅰ

1. Mechanics Exercise Class Ⅰ

2. 英文数字、算式表达法 = is equal to / equals >is larger than < is less than >> is much greater than a = b + c a is equal to b plus ca = b - c a is equal to b minus ca = b x c a is equal to b times c / a equals to b multiplied by ca = b/c a is equal to b divided by c / a equals to b over c (square) root x / the square root of x  cube root (of) x ;  fourth root (of) x ;   nth root (of) x 

3. 英文数字、算式表达法 f( x )    fx / f of x / the function f of x   the limit as x approaches zero   the integral from zero to infinity   x2x square / x to the second power / x to the power twox3x cube / x to the third power / x to the power three x7the seventh power of x（x to the seventh power） lognXlog x to the base n

4. Brief Review Displacement Velocity Acceleration Projectile Motion (trajectory) Centripetal Acceleration Newton’s Second Law Drag Force

5. 1. A model rocket fired vertically from the ground ascends with a constant vertical acceleration of the 4.00m/s2for 6.00 s. Its fuel is then exhausted ,so it continues upward as a free-fall particle and then falls back down. (a) What is the maximum altitude reached? (b) What is the total time elapsed from the takeoff until the rocket strikes the ground? Solution: (a) One key idea is since the fuel is exhausted and before the rocket strikes the ground, its acceleration is g of magnitude. And when the fuel is exhausted ,the velocity is (upward) y then we can get the position of the rocket 0

6. Second key idea is the velocity equals zero, when the rocket is at maximum altitude. So So the maximum altitude is (b) Since the fuel is exhausted and before the rocket returns, the time interval is Then the rocket from the maximum altitude falls back to the ground. From the Eq. 2-50, we can obtain

7. Work out this equation, yielding So the total time elapsed from the takeoff until the rocket strikes the ground is

8. T m1 m2 m2 m2g 2. A block of mass m1 on a frictionless inclined plane of angle is connected by a cord over a massless , frictionless pulley to a second block of mass m2 hanging vertically. What (a) the magnitude of the acceleration of each block and (b) The direction of the acceleration of the hanging block? (c) what is the tension of the cord? Solution: Choose m2 to be a system and draw it’s free-body diagram With Newton’s second law applied to the m2 system, we can obtain (1)

9. m1g N T Choose m1 to be a system and draw it’s free-body diagram and we can write Parallel direction Perpendicular direction The acceleration components a1 and a2, have the same value since the string does not stretch, thus (2) Now we can solve equation (1) and (2) simultaneously for T and a2. First solve equation (1) for T (3) Then substitute for T into equation (2): Solving for a2, we find

10. Discussion : If the direction of the acceleration of the hanging block is vertically downward. On the other hand if the direction is vertically upward. Substituting for a2 for Eq. (3), we find that tension in the rope is of magnitude

11. 3 A truck moves along a straight road with a constant speed of 30m/s, and a projectile is launched from it. What is the (a) magnitude and (b) direction of the projectile’s initial velocity relative to the truck if it can fall back to the launching point on the truck after the truck moves 60m. The drag force is negligible. Solution: Choose the truck as reference. Set the launching point as origin, and construct coordinates O-xy as shown. Set the launching time as starting point of time.

12. The loading time: And at the loading time so (1) (2) Eq (1) yields Eq (2) yields So, if the projectile is thrown vertically relative to the truck, it will fall back to the launching point after the truck moves 60m.

13. 4 An inclined plane of mass m2 can slide on a smooth horizontal surface, with the inclined angle of α. An athlete of mass m1 slides on the inclined plane frictionlessly, what is (a) the relative acceleration of the athlete with respect to the inclined plane and (b) the pressure force on the plane from the athlete? Solution：Isolating the bodies For For Solving the equations for a2 and N:

14. 5 A parachute athlete dives with an initial speed of v0, and the drag force is proportional to the square of the speed as: av2. The total mass of the parachute and the athlete is m. What is the function of v=v(t)? Hint: this can be utilized in the integration Solution: Transferring to

15. set the above equation can be written as Quadrature Then