00:01
Now in this question we are given that an object with weight w is dragged along a horizontal plane by a force acting along a rope attached to the object now the rope is making angle theta with the plane and the magnitude of this force if is given by mu times w divide with mu times sine theta plus cosine theta now this mu is coefficient of friction and theta ranges from 0 to pi by 2 and we need to show that f is minimized when then theta is equals to mu this is what we have to show now for that we have to find a differentiation of this function with respect to theta and then we'll go to evaluate it with zero that is first we'll find f prime theta and then will equate it to zero so f prime theta or you could say d f over d theta it will be equals to now you can do one thing d over d theta now this force we can write this value as mu times w multiplied with we can take this denominator into the numerator so it will become mu sine theta plus cosine theta to the power minus one now to differentiate this with respect to theta we have to use the chain rule here so mu times w this minus one would come into the front then minus one minus one that would become minus two so this would be mu times sine theta plus cosine theta to the power minus two multiplied with the differentiation of this expression.
02:08
So, mu as it is, now differentiation of sine theta is cosine theta, and differentiation of cosine theta is minus sine theta.
02:18
So that will be minus sine theta.
02:23
Now, how we can further simplify this? so this would be this minus sign as it is, mu w.
02:30
This value i'm writing here, so mu, cosine theta, minus sine theta, divide with this one.
02:40
Because we can take this one to the denominator.
02:42
So this is mu, sine theta plus cosine theta, and it's squared.
02:50
So this is f prime theta.
02:54
Now we're going to equate it with zero.
02:57
So it will be equals to...