Sec_1.2 - 1.2 Some Mathematic Models Ex1 Free Falling Body...

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1.2 Some Mathematic Models Ex1. Free Falling Body It is well known that free-falling objects close to the earth accelerate at a constant rate g . Assume s is a distant function of time, t , and upward is positive. Then, 2 2 . ds d s v g dt dt = - = - Ex2. Vibration of a Mass on a Spring To find the displacement ( 29 x t attached to a spring, we use Newton’s 2 nd law of motion and Hooke’s law. 1. Newton’s 2 nd law of motion: The net force acting on the system in motion is 2 2 d x F ma m dt = = 2. Hooke’s law: F k x = - , where k is a constant. From the above, it can be seen that 2 2 2 2 0 d x d x k m kx x dt dt m = - + = Define 2 k x ϖ = , then we have the differential equation
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Unformatted text preview: 2 k x ϖ = , then we have the differential equation 2 2 2 2 2 d x k d x x x dt m dt + = + = . Ex3. Simple Pendulum ***The picture is on page 14*** Again, the displacement s is a function of time and 2 2 2 2 d s d s l a l dt dt θ = = = . From the Newton’s 2 nd law, we have ( 29 ( 29 2 2 2 2 sin sin 0. d d g F ma m l mg dt dt l = = = -+ = When is small, ( 29 sin . So, the above ODE can changed to ( 29 2 2 2 2 sin d g d g dt l dt l + = + = . Ex4. Charge on a Capacitor...
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