GATE Mechanical 2017 Set 1 | GA | Question: 3

Question

A right-angled cone (with base radius $5$ cm and height $12$ cm), as shown in the figure below, is rolled on the ground keeping the point $P$ fixed until the point $Q$ (at the base of the cone, as shown) touches the ground again.

By what angle (in radians) about $P$ does the cone travel?

• A. $\dfrac{5\pi}{12} \\$
• B. $\dfrac{5\pi}{24} \\$
• C. $\dfrac{24\pi}{5} \\$
• D. $\dfrac{10\pi}{13}$

Comments

option d.

it will sweep horizontal distance equal to circumfrence=2*pi*r=2*pi*5=10pi

now, distance of point Q from P(using pythagoras)=sqrt(12^2 + 5^2)=13..

now during it course of journey, it will form circular arc, where

'P' being center of arc, '10pi' being length of arc and '13' being radius of circular arc..

now using formula for circular arc i.e

theta = l/r

theta = 10pi/13...

i will suggest you first draw a proper diagram, then you will get exactly what i have done..

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The original question was:

A right-angled cone (with base radius 55 cm and height 1212 cm), as shown in the figure below, is rolled on the ground keeping the point PP fixed until the point QQ (at the base of the cone, as shown) touches the ground again.

By what angle (in radians) about PP does the cone travel?

 

So you're saying that this is same as opening the cone?

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@Sambhrant Maurya

So you're saying that this is same as opening the cone?

This is correct, like this: If you will roll the cone then also you will get this circular arc as path of the cone and if you unroll the cone from the point $Q$ then also you will get this circular arc.

Answer 1

Point $Q$ will touch again, when cone will roll around $P$ and will travel arc length of $2\pi r$

So, arc length made by cone $= 2 \pi r = 10\pi $ cm

If the cone rotates one round around point $P$ it will cover perimeter of length $2 \pi l$ cm

where, $l = \sqrt{h^2+r^2} = 13$ cm

So, perimeter of one rotation $= 26 \pi$ cm

Thus, the angle(in radian) which cone makes $= \frac{10\pi}{26\pi}\times 360\times \frac{\pi}{180} = \frac{10\pi}{13}$

Correct Answer: $D$

Comments

@Naveen Kumar 3

Explain this part

 

Thus, the angle(in radian) which cone makes =10π/26π×360×π/180=10π/13

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@jlimbasiya, when cone will rotate one round( $2\pi l=26\pi$ distance) around $P$, then it will make $360^\circ$.Here, when point $Q$ again touches ground, it covers $10\pi$ arc length.

So, angle made by it = $\frac{10\pi}{26\pi}*360^\circ$   (this is in degree)

so, convert this result by multiplying with $\frac{\pi}{180}$ to convert it in radians.

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We can also directly multiply by 2π. Last second line in best answer

= (10π / 26 ) * 2 π

= 10 π / 13

So no confusion of degree and radian conversion and all that.

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@Kiyoshi

correct lines are:

$ = \Big( \dfrac{10 \pi}{26 \pi} \Big) \times 2 \pi $

$ = \dfrac{10 \pi}{13 \pi} $

PS: Basically, we want to know if whole circle or arc of length $ = 2 \pi r$ makes angle of $2 \pi$ on center then if we have a fraction of $2 \pi r$ how that angle will change. that's why we are taking fraction of $\dfrac{10 \pi}{26 \pi}$ that represent $\%$ length of circumference that the arc carries so it will be reflected on the angle made by that arc on center and that's why $2 \pi$ is multiplied here.

PS: PS: 

Q: Why does a complete circle (arc length $2 \pi r$) create an angle of $2 \pi$ radians at the center?

Ans: This relationship stems from the definition of a radian:

1. 1 radian is the angle at the center when the arc length equals the radius (r).
2. This creates a linear relationship:
   • 1 radian : arc length $= r$
   • 2 radians : arc length $= 2r$
   • And so on...
3. For a full circle:
   • Circumference $= 2 \pi r$**
   • Therefore, angle $= 2 \pi$ radians

**The value of $\pi$ was discovered through practical measurements, those experiments showed that for any circle, the ratio of its circumference to its diameter is always the same constant, which we call $\pi$. So, circumference formula probably came earlier then mathematicians thought of inventing radians as a unit of angle measurement.

Answer 2

Base radius = 5cm, Height = 12m. So Slant Height = $\sqrt{5^{2}+12^{2}}$ = 13cm.

We open it now. I appears as an arc if opened. Radius of the arc = Slant Height of the cone = 13cm.

Length of the arc = Circumfernce of the cone = 2*π*5 = 10πcm

Now, when, length of arc is 2*π*13, angle subtended is 2π.

So, angle subtended when length of arc is 10π = $\frac{2π}{26π} * 10π$ = $\frac{10π}{13}$

Comments

Thankyou it was a clear explanation!

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But can you tell me why atlast you multiplied it with 10π?

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@Aadhyaa Gupttaa

he used unitery method:

$\because$ If length of arc is $2\pi \times13$, angle subtended = $2 \pi$

$\therefore$ If length of arc is 1, angle subtended = $\dfrac{2 \pi}{2\pi \times13}$

$\therefore$ If length of arc is $10 \pi$, angle subtended = $\dfrac{2 \pi}{2\pi \times13} \times 10 \pi = \dfrac{10 \pi}{13}$

Answer 3

The Question is asking angle about P that the cone rolls.
When the cone is revolving around a fixed point P , it traces a circle.
The radius of the circle is Slant height of the cone and the arc length covered is the perimeter of the base of the cone.
We are asked when the cone completes one full rotation , what will be the angle formed at the center.

 

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arc length = radius  X angle(in radian)

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Here,
radius = Slant height
arc lenght = perimeter of base circle of cone
slant height = 13 (apply pythagoras theorem)
perimeter of base circle = 10pi
therefore angle = 10pi/13