Senim Silla
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Homework Statement
A model yacht runs on a horizontal frictionless oval track as shown (viewed from above) in the figure. The curved parts of the track are semi-circles of radius ##R = 0.5 m##; the straight sides have length ##L = 1 m##. The mass of the yacht is ##m = 0.5 kg.##
A force of magnitude ##|F| = 4 N## is applied to the sails, using a fan as shown. The force is horizontal, directed at π/6 to the direction of the straight track. Both the magnitude and direction of the force remain constant throughout the game.
(a) After passing B the yacht enters the curved section of the track on the right-hand side of the diagram. Find the maximum speed reached by the yacht on this curved section, and the angle θ (defined as shown) at which this maximum speed is attained.
(b) Will the yacht travel all the way around the track and back to point A before it comes to a halt? If not, where will it stop?
Homework Equations
The Attempt at a Solution
When I approached this problem, I thought about the centripetal force provided by the reaction force from the track on the yacht, pointing inwards to the center of one of the semi-circles. I thought that this central force would not be constant as the yacht goes around the bend, and that it would be:
$$F = 4\sin30\sinθ \hat i + 4\cos30\cosθ \hat j $$
The fact that it is not constant is what confuses me when trying to solve the question. I am not sure how to apply equations that I am familiar with, eg.
$$ m(\ddot{r}-r\ddot{θ}) = F(r) $$
$$ v = r\dot{θ}$$
While I calculated the speed of the yacht when it reaches B (3.7m/s), I am not sure how to follow through.
(The given answers are: The maximum speed will be when θ = 2π/3, and is vmax = 5.08 m/s. The yacht does not make it back to point A; it stops a third of the way round the left-hand curve).
Any help appreciated, thanks
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