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Unformatted text preview: PHYS 230 Fall 2007 Assignment 4 1. A continuation of the mass-and-spring on a cart problem of assignment 2. You may use the results from assignment 2. Textbook: pages 34/5 and 55/56. Your own class notes: end of class 7. 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) In the accelerating frame, what is the equation that describes the motion of the mass? Illustrate with a diagram of displacement from equilibrium versus time. Take the displacement x to be positive to the right, in the direction of the positive coordinate axis. The mass-spring system will oscillate. To figure out the details of the oscillation, follow the same steps as in assignment 2: choose the origin of coordinates as the left-hand end of the unstretched spring. Then the equilibrium position is with the spring extended by x =- . 5 m, the angular frequency is = q ( k/m ) = 2 s- 1 , and the amplitude is | x | = 0 . 5 m. The equilibrium point is to the left of the position at t = 0 . Or, at t = 0 , the displacement from equilibrium is to the right. Thus, if we write x- x = A cos ( t + ) , we have 0 + 0 . 5 = 0 . 5 cos (2 0 + ) , so that = 0 , and x + 0 . 5 = 0 . 5 cos 2 t (b) In the inertial frame of an observer who is not accelerating with the platform, what is the equation that describes the motion of the mass? Illustrate with a diagram of displacement from the initial position versus time. The motion of the platform, and therefore of the origin of coordinates on the platform, seen by an observer watching the platform, is x = X + 1 2 at 2 , where X is a constant, which we can set to zero. (Note: we choose the frame of the platform to be S , and the frame of the observer to be S .) Therefore, the motion of the mass as seen by the observer is x = 1 2 at 2 + x = t 2 + 0 . 5(cos 2 t- 1) 1 1 2 3 4 5 6 10 20 30 40 time displacement Here is the graph: the motion is dominated by the acceleration of the platform, which is the thin line. 2. From last years final exam. Textbook: pages 34/5 and 55/6. Your own course notes: class 7. Small quantities: notes on WebCT. A pendulum is constructed from a (massless) T-shaped frame carrying three equal masses at its corners. It hangs from the centre of its shortest side, as shown. Its angular displacement is defined with respect to the vertical equilibrium position, as shown. The angle is small....
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This note was uploaded on 04/29/2008 for the course PHYS 230 taught by Professor Harris during the Fall '07 term at McGill.
- Fall '07