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Full derivation of SHO around the equilibrium point of a curve in a constant gravitational field in the absence of friction

Full derivation of SHO around the equilibrium point of a curve in a constant gravitational field in the absence of friction

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help me to do part a; this is not a graded quesiton btw

Full derivation of SHO around the equilibrium point of a curve in a constant gravitational field in the absence of friction Let's recall the main equation used to derive the motion of a point particle along a curve in the presence of constant gravity without friction: 1/2 m (v² + v²) +mgy = E, (5) where m, g and E are constants, x is the coordinate along the horizontal axis, y is a smooth function of rand va, vy are the velocity components along each axis. Since the object is constrained to move along the curve, y-y(x), the components of the velocity satisfy vydy/dt = (dy/dx) (dx/dt)= ve y'. Let's assume that there is an equilibrium point at (x, y) = (xo, yo). (a) In the following, let's consider the motion of the particle within a range of x around to such that there isn't any local maximum of y(x) within this range. Describe with words and figures the kind of situations that this condition avoids. We move the particle up to a height H and, from rest, we let it slide. (b) What's E in terms of the parameters of the problem? Hint: easy. (c) Derive an expression for Eq. 5 that contains the variables va and Ax = x - xo only, including terms up to (Ar)². Collect terms with higher order corrections symbolically with the notation O((Ax)³) Hint: besides v, and Ax, you'll get some coefficients that depend on y(x) and its derivatives at xo. Make use of the fact that xo is an equilibrium point to simplify some of the terms. (d) Prove that the resulting equation corresponds to SHO, and show that the equiv- alent spring stiffness, k, has two terms: one of them is constant and the other one is proportional to both, the maximum change of height that the particle can undergo, and the value of the second derivative at the equilibrium point. Hint: check that units make sense to avoid any typos. (e) Show that the second derivative at the equilibrium point can be related to the radius of curvature of y(x) at the same point Since we are considering small oscillations, prove that the final expression of Eq. 5 is the one shown in class: 1 [m² + 1 mgy (Ax)² = mg (H - yo). Where we have neglected terms of order (Ax)³ and higher. This result proves that indeed there is SHO around the equilibrium point of a curve in a constant gravitational field in the absence of friction. There's one more question. 6Consider y(ar) a C function until the very last question. Big O Notation *For a definition of radius of curvature, see Wikipedia: radius of curvature
Transcribed Image Text:Full derivation of SHO around the equilibrium point of a curve in a constant gravitational field in the absence of friction Let's recall the main equation used to derive the motion of a point particle along a curve in the presence of constant gravity without friction: 1/2 m (v² + v²) +mgy = E, (5) where m, g and E are constants, x is the coordinate along the horizontal axis, y is a smooth function of rand va, vy are the velocity components along each axis. Since the object is constrained to move along the curve, y-y(x), the components of the velocity satisfy vydy/dt = (dy/dx) (dx/dt)= ve y'. Let's assume that there is an equilibrium point at (x, y) = (xo, yo). (a) In the following, let's consider the motion of the particle within a range of x around to such that there isn't any local maximum of y(x) within this range. Describe with words and figures the kind of situations that this condition avoids. We move the particle up to a height H and, from rest, we let it slide. (b) What's E in terms of the parameters of the problem? Hint: easy. (c) Derive an expression for Eq. 5 that contains the variables va and Ax = x - xo only, including terms up to (Ar)². Collect terms with higher order corrections symbolically with the notation O((Ax)³) Hint: besides v, and Ax, you'll get some coefficients that depend on y(x) and its derivatives at xo. Make use of the fact that xo is an equilibrium point to simplify some of the terms. (d) Prove that the resulting equation corresponds to SHO, and show that the equiv- alent spring stiffness, k, has two terms: one of them is constant and the other one is proportional to both, the maximum change of height that the particle can undergo, and the value of the second derivative at the equilibrium point. Hint: check that units make sense to avoid any typos. (e) Show that the second derivative at the equilibrium point can be related to the radius of curvature of y(x) at the same point Since we are considering small oscillations, prove that the final expression of Eq. 5 is the one shown in class: 1 [m² + 1 mgy (Ax)² = mg (H - yo). Where we have neglected terms of order (Ax)³ and higher. This result proves that indeed there is SHO around the equilibrium point of a curve in a constant gravitational field in the absence of friction. There's one more question. 6Consider y(ar) a C function until the very last question. Big O Notation *For a definition of radius of curvature, see Wikipedia: radius of curvature
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