Question

Applying Newton's 2nd law, Fnet=ma, then to this system, along the direction of motion (parallel to the swinging bob), we find −mgsin⁡θ=ma. (Note, the negative sign is required since the acceleration in our picture, which is to the right at this moment, is opposite to the angular displacement from the vertical, which is to the left.)

This equation is extremely difficult to solve, so let us simplify it by assuming that the pendulum's swings only through a very small angle, and therefore, the motion is primarily along the horizontal. Therefore, sin⁡θ=xl, where x is the position of the pendulum measured with respect to an origin located at the center of the bob's swinging motion. Also, as the motion is assumed to be approximately on the horizontal, $LaTeX: a=\frac{d^2 x}{dt^2}$ a=d2xdt2, by definition. Hence, Newton's 2nd law for this system becomes −mglx=md2xdt2. This "linear differential" equation is much easier to solve for the position x at any time, even if you haven't studied differential equations. Can you think of a mathematical function that when you take the derivative of it twice, the solution is the negative of that function.

net ma -mg sin θ-ma sin θ mg dt2 Question 2 0.33 pts Applying Newton's 2nd law, Fnet ma, then to this system, along the direction of motion (parallel to the swinging bob), we find -mgsin8 -ma. (Note, the negative sign is required since the acceleration in our picture, which is to the right at this moment, is opposite to the angular displacement from the vertical, which is to the left.) This equation is extremely difficult to solve, so let us simplify it by assuming that the pendulum's swings only through a very small angle, and therefore, the motion is primarily along the horizontal Therefore, sin θ where z is the position of the pendulum measured with respect to an origin located at the center of the bob's swinging motion. Also, as the motion is assumed to be approximately on the horizontal, a da by defnition. Hence, Newton's 2nd law for this system dt2 m.g This "linear differential" equation is much easier to solve for the becomes-^m-. position r at any time, even if you haven't studied differential equations. Can you think of a mathematical function that when you take the derivative of it twice, the solution is the negative of that function dt2 square roots squaring O logs O cosines or sines

Answer 1

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