This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Step 1: Start with the given displacement equation for simple harmonic motion. The displacement x of a simple oscillator as a function of time t is given by: x = x_0 ( t) where x_0 is the amplitude and is the angular frequency. Step 2: Differentiate the displacement equation with respect to time to find the velocity. Velocity v is the first derivative of displacement with respect to time: v = (dx)/(dt) v = (d)/(dt) (x_0 ( t)) Using the chain rule, (d)/(dt) ((at)) = a (at): v = x_0 ( t) Step 3: Differentiate the velocity equation with respect to time to find the acceleration. Acceleration a is the first derivative of velocity with respect to time (or the second derivative of displacement): a = (dv)/(dt) a = (d)/(dt) (x_0 ( t)) Using the chain rule, (d)/(dt) ((at)) = -a (at): a = x_0 (- ( t)) a = -x_0 ^2 ( t) Step 4: Substitute the expression for x back into the acceleration equation. From Step 1, we know that x = x_0 ( t). We can substitute this into the expression for a from Step 3: a = -^2 (x_0 ( t)) a = -^2 x This proves that for simple oscillations, the acceleration is directly proportional to the negative of the displacement, which is the defining characteristic of simple harmonic motion. The final expression for acceleration is: a = -^2 x Send me the next one 📸