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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: m (v² + v₁) +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 ve, 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

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: m (v² + v₁) +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 ve, 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

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

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 (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 ve, 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 vy= dy/dt = (dy/dx) (dx/dt) = vy. 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 v 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 ro. Make use of the fact that zo is an equilibrium point to simplify some of the terms.
Transcribed Image Text: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 (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 ve, 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 vy= dy/dt = (dy/dx) (dx/dt) = vy. 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 v 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 ro. Make use of the fact that zo is an equilibrium point to simplify some of the terms.
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