Assignment 2

Assignment 2 - PHYS 230 Fall 2007 Assignment 2 This...

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PHYS 230 Fall 2007 Assignment 2 This assignment is about accelerated (non-inertial) reference frames. Unless otherwise specified, solve these problems in the appropriate accelerated frame. 1. [5 points.] A boy is riding on a flat rail-car, on level ground, that has an acceleration a in the direction of its motion. (It is speeding up!) At what angle to the vertical should he toss a ball so that he can catch it without shifting his position on the car? Is the angle forwards or backwards? In the accelerated frame of the rail-car, the forces on the ball will be mg , downwards, and - ma , in the direction opposite to the acceleration a . This is as if there is an effective gravitational force, whose value is m g 2 + a 2 , and whose angle to the vertical is arctan a/g . The ball will therefore “return” directly to the boy if he throws it ex- actly opposite to the direction of this force. (Think of throwing a ball upwards against “normal” gravity!) 2. [5 points.] The “conventional” way to solve this problem takes much longer. Suppose that the intial velocity of the rail-car is u . Suppose that the ball has initial vertical velocity v y and horizontal velocity (which does not change) v x + u . Then, the time to return to the ground (level of the rail-car) is t = 2 v y /g . In this time, the horizontal distance traveled by the ball is d = ( v x + u ) t = 2 v y ( v x + u ) /g . In this time t , however, the rail-car and boy do not necessarily travel the same distance as the ball. They travel distance D = ut + 1 2 at 2 . Thus, the condition that we want is d = D , or 2 v y ( v x + u ) /g = 2 v y u/g + 1 2 a (2 v y /g ) 2 . This simplifies to give a/g = v x /v y as required. 3. [5 points.] A block of mass 2 kg rests on a frictionless platform on wheels. It is attached to a horizontal spring of spring constant 8 N/m, as shown in the figure. Initially the whole system is stationary, but at t = 0 the platform begins to move to the right with a constant acceleration of 2 m/sec 2 . As a result, the block begins to oscillate horizontally relative to the platform. - 2 m/s 2 d d 2 kg 8 N/m (a) What would be the new equilibrium extension of the spring? Imagine that all the oscillations have died away, but that the platform is still accelerating. The inertial force on the mass is
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Assignment 2 - PHYS 230 Fall 2007 Assignment 2 This...

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