A pin is forced to move along a parabola shaped guide by a connected collar that

Question

A pin is forced to move along a parabola shaped guide by a connected collar that moves along the x-axis only. When x = 1.3 m the velocity and acceleration of the collar are: v_x = -1 m/s and a_x = 2.7 m/s^2 respectively. The parabolic guide is described by y = -1.25x^2 + 5x, where both x and y are in units of metres. (a) What is the y-component of velocity, v_y at this instant? (b) What is the magnitude of the acceleration at this instant? (c) What is the normal component of the acceleration at this instant?

Explanation / Answer

Given,

Position, x = 1.3 m

Velocity, vx = -1 m/s

Acceleration, ax = 2.7 m/s2

(a) vy = dy/dt = -2.5 x vx + 5vx

At x = 1.3 m

vy = (-2.5 x 1.3 x (-1)) + (5 x (-1)) = -1.75 m/s

(b) ay = dvy/dt = -2.5(vx2 + xax) + 5ax

At x = 1.3 m,

ay = -2.5[(-1)2 + (1.3 x 2.7)] + (5 x 2.7) = 2.225 m/s2

Magnitude of acceleration, a = (ax2 + ay2)1/2 = [2.72 + 2.2252]1/2 = 3.50 m/s2

(c) Normal component of acceleration, ay = 2.225 m/s2

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