# Calculate the minimum force required

In summary: misconception... to think that the block will always move when there is a force that is greater than the friction.
PeroK said:
You need calculus to get that answer. It's a minimization problem. That generally means calculus is required.
I understand your point but that's how the question is given
anyways thank you for giving me your precious time

PeroK
PeroK said:
Generally, you need calculus to minimise something in a problem with a continuous variable.
But not always, and not here.
@ALIAHMAD , if you write the two force balance equations and eliminate the normal force you should end up with ##F(\cos(\theta)+\mu\sin(\theta))=\mu W##.
The trick then is to multiply by a constant such that the trig terms can be collapsed into one: ##F(A\cos(\theta)+A\mu\sin(\theta))=A\mu W## where ##A=\sin(\alpha), A\mu=\cos(\alpha)##. Can you see how to choose A such that ##\alpha## exists? Can you then collapse that into a single trig term, and then spot what value of ##\theta## minimises F?

Orodruin and SammyS
haruspex said:
But not always, and not here.
@ALIAHMAD , if you write the two force balance equations and eliminate the normal force you should end up with ##F(\cos(\theta)+\mu\sin(\theta))=\mu W##.
The trick then is to multiply by a constant such that the trig terms can be collapsed into one: ##F(A\cos(\theta)+A\mu\sin(\theta))=A\mu W## where ##A=\sin(\alpha), A\mu=\cos(\alpha)##. Can you see how to choose A such that ##\alpha## exists? Can you then collapse that into a single trig term, and then spot what value of ##\theta## minimises F?
yeah I can do these
like the combined trig term is sin(θ+𝜶)
my teacher told me that considering θ will make the question more complicated as for every angle there is some minimum and then to find the minimalist of them is very hard
he told me that these types of questions are solved by using this formula

F(min)=mgμ/√(1+μ^2)

yeah I can do these
like the combined trig term is sin(θ+𝜶)
my teacher told me that considering θ will make the question more complicated as for every angle there is some minimum and then to find the minimalist of them is very hard
he told me that these types of questions are solved by using this formula

F(min)=mgμ/√(1+μ^2)
Or, by deriving that formula.

the combined trig term is sin(θ+𝜶)
Good, so what is the equation you get for F and what is A in terms of ##\mu##?

haruspex said:
Good, so what is the equation you get for F and what is A in terms of ##\mu##?
F=Cos𝜶W/[sin(θ+𝜶)] and A= cos𝜶/μ

just wanted to know how that formula is derived as I don't see any clue on it

F=Cos𝜶W/[sin(θ+𝜶)] and A= cos𝜶/μ
I meant, find A purely in terms of ##\mu##. ##\alpha## should not appear. But forget that, and instead find ##\alpha## purely in terms of ##\mu##.
Use the equations ##A=\sin(\alpha), A\mu=\cos(\alpha)## and eliminate A.
Given your first equation above, what value of ##\theta## minimises F?

F=Cos𝜶W/[sin(θ+𝜶)] and A= cos𝜶/μ

just wanted to know how that formula is derived as I don't see any clue on it
It seems you have not really gotten the point here. The point is to introduce a constant ##A##, expressed only in the variable ##\mu##, that allows you to rewrite your expression using a single trigonometric function with ##\theta+\alpha## as its argument.

Edit: Please note that it is also beneficial for you to write out the entire argument rather than only a result as people will be able to help you better when you go wrong.

PeroK
haruspex said:
But not always, and not here.
@ALIAHMAD , if you write the two force balance equations and eliminate the normal force you should end up with ##F(\cos(\theta)+\mu\sin(\theta))=\mu W##.
The trick then is to multiply by a constant such that the trig terms can be collapsed into one: ##F(A\cos(\theta)+A\mu\sin(\theta))=A\mu W## where ##A=\sin(\alpha), A\mu=\cos(\alpha)##. Can you see how to choose A such that ##\alpha## exists? Can you then collapse that into a single trig term, and then spot what value of ##\theta## minimises F?
Here's a simpler alternative. To minimise ##F## we need to maximise ##\cos(\theta)+\mu\sin(\theta)##. That's easy once you know some calculus.

Failing that, we let ##\mu = \tan \alpha## (for some ##0 < \alpha < \frac \pi 2)## and aim to maximise ##\cos(\theta)+\tan(\alpha) \sin(\theta)##. One more tweak and a trig identity should do it.

PeroK said:
That's easy once you know some calculus.
Sure, but the point of my post was, in view of post #28, to show it can be done without any calculus. Besides, it is a useful trick in another context: solving equations involving a mix of trig functions of the same unknown.

haruspex said:
Sure, but the point of my post was, in view of post #28, to show it can be done without any calculus. Besides, it is a useful trick in another context: solving equations involving a mix of trig functions of the same unknown.
I get that, but a little differentiation goes a long way!

Firstly you should draw a free body diagram . Check what is asked , here you need MINIMUM force so just enough to move the block but not enough to accelerate it . So Force appliead is equal to friction applied . Now how do you know which angle to apply the force on ? Perpendicular? Parallel? Well for now we have no clue so we make the mathematical equations considering it was applied at an angle *theta* and if minimum force was applied Parallel to block the theta would turn out to be zero, the math will do the work for you , all you have to do is arrive at the proper equations . I assume you have been taught some trigonometry as you are solving questions from NLM [A side note :- if you are struggling with step 2 questions i suggest you solve the solved examples then go for step-1 , maybe solve step-3 examples after exercise, they help with concept building :)

You will face some differentiation and integration while solving PYQ , and especially whlie solving advanced questions so it is better to know some basic stuff , when to use it and when to use which (diff or integration) and how to use it.

Spector989 said:
So Force appliead is equal to friction applied
This is incorrect.
As has already been discussed in the thread, friction must be equal to the horizontal component of the applied force.
Spector989 said:
You will face some differentiation and integration while solving PYQ , and especially whlie solving advanced questions so it is better to know some basic stuff , when to use it and when to use which (diff or integration) and how to use it.
That’s fine but — again as already discussed in the thread — differentiation is not necessary in this problem.

Orodruin said:
This is incorrect.
As has already been discussed in the thread, friction must be equal to the horizontal component of the applied force.

That’s fine but — again as already discussed in the thread — differentiation is not necessary in this problem.
Mb i should have mentioned it is equal to the horizontal component and Normal is equal to mg - minus vertical component (given force is applied above horizontal) and i simply siad that so that he wouldn't back off from differentiation and integration too much , in jee 11th you will face some calculus problems while solving some physics questions. But again it is my fault because i was not clear . Sorry