• Benjamin_harsh
In summary: The direction perpendicular to the surface is not affected by the incline because the block is sliding down an incline.
Benjamin_harsh
Homework Statement
A block of mass M is released from point P on a rough inclined
plane with inclination angle θ, shown in the figure below. The coeffucient
of friction is μ. if μ < tan θ, then time taken by the block to reach another
point Q on the inclined plane, where PQ = s, is
Relevant Equations
What is the meaning of " PQ = s" in the question?

How ##t = \large\sqrt\frac{2s}{a} ##?
Sol:

##Mg.sinθ - μMg.cosθ = ma##
##a = g.sinθ - μg.cosθ##

Now ##S = ut + \large\frac{1}{2}\normalsize at^{2}##

but ##u = 0##

##t = \large\sqrt\frac{2s}{a} = \sqrt\frac {2s}{gcosθ(tanθ- μ)}##

My questions:

What is the meaning of " PQ = s" in the question?

How ##t = \large\sqrt\frac{2s}{a} ##?

How ##a =gcosθ(tanθ- μ) ## in that last square root step?

s=a/2 t^2 is a standard formula, that's how he got a=sqrt (...)

a=g(sin theta - mu * cos theta) just comes from considering forces parallel and perpendicular to the plane where it slides. PQ=s just means that instead of always writing "PQ" he will write "s" because its shorter.

Explain this equation ##Mg.sinθ - μMg.cosθ = ma## ? I know standard sliding block equation: ##w·sinθ – μkN = 0## &

##N – w·cosθ = 0##How to understand ##μ < tan θ##?

Last edited:
Benjamin_harsh said:
Explain this equation ##Mg.sinθ - μMg.cosθ = ma## ? I know standard sliding block equation: ##w·sinθ – μkN = 0## &

##N – w·cosθ = 0##How to understand ##μ < tan θ##?
Yes you just take N=W cos theta and put it into te standard sliding block equation. Then you immediately get what you want

mu<tan(theta) because otherwise the block won't start sliding, the friction holds it in place. You can see that the equation of t=sqrt (...) gives you a complex number

Benjamin_harsh said:
I know standard sliding block equation: w⋅sinθ–μkN=0w·sinθ – μkN = 0
This equation is not standard for a block sliding down an incline with only gravity and incline forces acting on it. It is correct only if the acceleration of the block along the incline is zero.

Replusz
Lol you right mate. There is so much wrong with how he labels stuff i didnt even realize the =0 at the end

Yes, so its not 0. Its acceleration

No need of ## ΣF_{X} = 0## & ##ΣF_{y} = 0## for this sliding block problem? If yes, when to use ##ΣF_{X} = 0## & ##ΣF_{y} = 0## ?

Jesus.

N=Mk, where k is my acceleration, M is mass, and N is force. So now if we k=0 then sum(N)=0 but ONLY when k =0! But then it work good!

In this problem k not is 0 because you the acceleration have in parralel with the" slide ".

Benjamin_harsh said:
No need of ## ΣF_{X} = 0## & ##ΣF_{y} = 0## for this sliding block problem? If yes, when to use ##ΣF_{X} = 0## & ##ΣF_{y} = 0## ?
The general form for the two equations that is always correct to write down is
## ΣF_{x} = ma_x~;~ ~ΣF_{y} = ma_y## where ##a_x## and ##a_y## are the components of the acceleration. The sum of the forces in either direction is zero only if you have good reason to believe that it is. In this particular problem you have a block sliding down the incline along the ##y##-axis. This is enough reason to know something about the acceleration in the ##x##-direction. What is that? You are also told that the coefficient of (static) friction is less than ##\tan\theta.## That says something about the acceleration in the ##y##-direction. What could that be? Hint: What is the statement of the problem trying to tell you with ##\mu<\tan\theta##?

kuruman said:
The sum of the forces in either direction is zero only if you have good reason to believe that it is.

how can I tell The sum of the forces in either direction is zero ? Good reasons like what?

The net force in either direction is zero if the acceleration in that direction is zero. Of the two directions, perpendicular and parallel to the incline, is there a direction along which you can tell for sure that the acceleration is zero and hence the net force is also zero? Hint: Acceleration is a measure of change in velocity.

Benjamin_harsh
Acceleration if no friction exist; ##a_{x} = mg.sinθ##

Acceleration if friction exist; ##a = g.sinθ - µ_{k}.g.cosθ## (I am not sure whether it is ##μ_{s}## or ##µ_{k}## in this equation)

Are these equations correct for sliding block equations?

Benjamin_harsh said:
Acceleration if no friction exist; ##a_{x} = mg.sinθ##
[\quote]
Incorrect. There should be ##m## on the right side.
Benjamin_harsh said:
Acceleration if friction exist; ##a = g.sinθ - µ_{k}.g.cosθ## (I am not sure whether it is ##μ_{s}## or ##µ_{k}## in this equation)
This is correct but only if the gravity and the incline exert forces on the block. You should use ##\mu_k## if the block is moving relative to the surface. You may use ##\mu_s## in that equation if the block is at rest relative to the surface but only if the block is just on the verge to start sliding.
Benjamin_harsh said:
Are these equations correct for sliding block equations?
They are correct but subject to the conditions stated above for the direction parallel to the surface. What about the direction perpendicular to the surface? What can you say about that?

## 1. How do you calculate the acceleration of a block sliding down an inclined plane?

The acceleration of a block sliding down an inclined plane can be calculated using the formula a = g * sin(theta), where g is the acceleration due to gravity and theta is the angle of the inclined plane.

## 2. What factors affect the acceleration of a block sliding down an incline?

The acceleration of a block sliding down an incline is affected by the angle of the incline, the mass of the block, and the coefficient of friction between the block and the incline.

## 3. How does the angle of an inclined plane affect the speed of a sliding block?

The angle of an inclined plane affects the speed of a sliding block because it determines the component of the force of gravity that is acting parallel to the incline, which in turn affects the acceleration of the block.

## 4. What is the relationship between the coefficient of friction and the speed of a sliding block?

The coefficient of friction is a measure of the resistance between two surfaces in contact. As the coefficient of friction increases, the speed of a sliding block decreases because there is more resistance opposing its motion.

## 5. Can the acceleration of a sliding block ever be greater than the acceleration due to gravity?

No, the acceleration of a sliding block can never be greater than the acceleration due to gravity. This is because the force of gravity is always acting on the block, and any additional forces, such as friction, will only decrease the acceleration.

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