00:01
In this question, first let us draw a diagram of the piston which we are going to talk about and then we will understand it.
00:09
So, this is the piston that we have been given here.
00:17
This point is given as a, b and c.
00:20
This length is given to be 7 inches and this radius of this piston is given to be 2 inches.
00:29
Also, we have that angular velocity of b, it is omega b, i will write it to be, it is equals to 5 pi radian per second.
00:41
We have to find out the speed of the piston at the moment when the line bc, it makes an angle of pi over 4 with the horizontal.
00:53
At that time, what is the speed of the piston? that is what we have to find.
00:58
To find this, first of all, what we are going to do? we will apply the cosine law in the triangle abc, apply cosine rule in triangle abc, we get the following, ac square times ab square, sorry, ac square is equals to ab square plus bc square minus 2abbc cosine of theta.
01:37
So, ac square comes out to be ab is 7 plus 2 square minus 2 times 7 times 2 and in this case, cosine theta is just theta, correct? so, this comes out to be ac is equals to under root of 53 minus 28 cosine of theta.
02:02
We can find out the velocity of c.
02:07
So, it will be dac over dt, which is the velocity that is d over dt of under root 53 minus 28 cosine of theta.
02:19
Simplifying with this, we get 53 minus 28 cosine of theta minus 1 over 2 times, the derivative of this thing will come out to be 14 sin theta d theta over dt...