Question

A horse is tied to a post by a rope. If the horse moves along a circular path, always keeping the rope tight and describes 88 metres when it traces 72° at the centre, find the length of the rope.

Answer 1

Hint: Convert the given angle in degree to radians by multiplying it with $\dfrac{\pi }{{180}}$. Now draw the figure representing the same. Take the arc length as 88 metres and radius as $r$, as the horse is moving in a circular path. Substitute these values in the formula for angular displacement and get the value of $r$.

Complete step-by-step answer:

It is said that a horse is tied to a post by a rope. Let us consider the post as P. Now consider PA as the length of the rope to which the horse is tied. Let us assume that the horse moves along the arc AB so that $\angle APB=72{}^\circ$.

The length of the arc AB is given as 88 metres. From the figure you can understand more of it.

Let us take $r$as the length of the slope PA, i.e. $PA=r$.

It is said that the horse moves in a circular path. Hence we need to find the Angular displacement. The angle given is the shortest angle between the initial and final position of the horse.

Now let us convert the given angle of degrees to radians. To convert degrees to radians multiply the angle with $\dfrac{\pi }{180}$. Let us consider the angle as equal to $\theta$. Hence, $\theta =72{}^\circ$. Now let us convert it to radians.

& \theta =72{}^\circ =\left( 72\times \dfrac{\pi }{180} \right) \\\ & \theta =\dfrac{2\pi }{5} \\\ \end{aligned}$$ Thus we got the angle in radians as $$\dfrac{2\pi }{5}$$. We know the basic formula that, Angular displacement, $$\theta =\dfrac{arc}{radius}$$ We know the length of the arc as 88 metres and radius as $$r$$. Substitute these values in the equation of angular displacement. $$\theta =\dfrac{arc}{radius}$$ $$\dfrac{2\pi }{5}=\dfrac{88}{r}$$ Now let us apply the cross multiplication property and apply $$\pi =\dfrac{22}{7}$$. Thus we get,$$\begin{aligned} & r=\dfrac{88\times 5}{2\pi } \\\ & \Rightarrow r=\dfrac{88\times 5\times 7}{2\times 22}=\dfrac{88\times 5\times 7}{44} \\\ & \Rightarrow r=2\times 5\times 7=70 \\\ & \Rightarrow r=70m \\\ \end{aligned}$$ Hence we got the length of the slope PA as, $$r=70m$$. Hence the length of the rope to which the horse is tied is of $$70m$$ in length. Note: Don’t forget to convert the given angle into radians, as the angular displacement of an object is always given in radians. Angular displacement is the shortest angle between the initial and final position for a given object, which is having a circular motion.