The endpoints of a movable rod of length 1 meter have coordinates (x, 0) and (0,

Question

The endpoints of a movable rod of length 1 meter have coordinates (x, 0) and (0, y) (see figure). The position of the end on the x-axis is x(t) = ½ sin ut/6 where t is the time in seconds. Find the time of one complete cycle of the rod. What is the lowest point reached by the end of the rod on the y-axis? Find the speed of the y-axis endpoint when the x-axis endpoint is (1/4, 0). Repeat Exercise 35 for a position function of x(t) = sin Use the point (1/10, 0) for part (c).

Explanation / Answer

x(t) = 3/5 sin(pi*t) a) time, one complete cycle = 2pi/ pi = 2 sec b) lowest point occurs when x = +3/5 or -3/5 x^2 + y^2 = 1 => 9/25 + y^2 = 1 => y^2 = 16/25 y = 4/5 thus, point is y=4/5 c) speed of y-axis endpoint when x = 3/10 x^2 + y^2 = 1 => 2xdx/dt + 2ydy/dt = 0 => dy/dt = (-xdx/dt)/y and dx/dt = 3pi/5 cos(pi*t) => dy/dt = (-x* (3pi/5)*cost(pi*t))/y x(t) = 3/5 sin(pi*t), so when x = 3/10 so sin(pi*t) = 1/2 => pi*t = pi/6, 5pi/6 ... etc => t = 1/6 , 5/6 ... etc note, y = sqrt(91)/10 when x=3/10 at x= 3/10 and t = 1/6 dy/dt = (- 3/10 * 3pi/5 * cos(pi/6) )/ sqrt(91)/10 = - 9 sqrt(3)/ sqrt(91) m/s at x= 3/10 and t = 5/6 dy/dt = + 9 sqrt(3)/ sqrt(91) m/s hence, speed of y-axis endpoint when x = 3/10 is 9 sqrt(3)/ sqrt(91) m/s

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