# How do you calculate the acceleration of a block sliding down a plane?

• tryingtolearn1
In summary: Yes, that I understand but what does it mean physically for the block to travel a horizontal direction? The block is inclined so the only direction it can physically travel is in the direction of the incline so why is the...
tryingtolearn1
Homework Statement
A block starts at rest and slides down a frictionless plane inclined at an angle $\theta$. What should $\theta$ be so that the block travels a given horizontal distance in the minimum amount of time?
Relevant Equations
$$F=ma$$
When drawing a diagram of the forces acting on the block, I have the following forces: $$\sum F_x =mg\sin\theta = ma$$ $$g\sin\theta = a$$ however the solution has $$F_x = ax = (g \sin\theta) \cos \theta$$ but I am not sure how they got that? I know the normal force is $$N=mg\cos\theta$$ but the normal acts on the y-axis.

tryingtolearn1 said:
Homework Statement:: A block starts at rest and slides down a frictionless plane inclined at an angle $\theta$. What should $\theta$ be so that the block travels a given horizontal distance in the minimum amount of time?
Relevant Equations:: $$F=ma$$

When drawing a diagram of the forces acting on the block, I have the following forces: $$\sum F_x =mg\sin\theta = ma$$ $$g\sin\theta = a$$ however the solution has $$F_x = ax = (g \sin\theta) \cos \theta$$ but I am not sure how they got that? I know the normal force is $$N=mg\cos\theta$$ but the normal acts on the y-axis.
Are you using the same XY axes as in the solution?

haruspex said:
Are you using the same XY axes as in the solution?

yes, I am using the x-y plane to determine the acceleration. The book states the following:

The component of gravity along the plane is $$g \sin\theta$$. The acceleration in the horizontal direction is therefore $$a_x = (g \sin\theta) \cos\theta$$

tryingtolearn1 said:
yes, I am using the x-y plane to determine the acceleration. The book states the following:
That's not what I asked. My question is whether your XY axes are the same as in the official solution.
Looks to me that the book takes X as horizontal and Y as vertical , whereas you are taking X as parallel to the plane and Y as normal to the plane. That would explain the differences in the algebra.

haruspex said:
That's not what I asked. My question is whether your XY axes are the same as in the official solution.
Looks to me that the book takes X as horizontal and Y as vertical , whereas you are taking X as parallel to the plane and Y as normal to the plane. That would explain the differences in the algebra.

The solution doesn't provide a diagram so I am not sure what XY axis my books takes. Here is the diagram I drew that I assumed is what the book claimed:

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tryingtolearn1 said:
The solution doesn't provide a diagram so I am not sure what XY axis my books takes. Here is the diagram I drew that I assumed is what the book claimed:
So rework it using X as horizontal and Y as vertical and see whether you get the same as the book.

haruspex said:
So rework it using X as horizontal and Y as vertical and see whether you get the same as the book.

Hm, I don't understand? Isn't this problem exactly similar to finding the acceleration of a block sliding down on a frictionless inclined plane thus the acceleration is $$a=g\sin\theta?$$
Which can be found here
http://hyperphysics.phy-astr.gsu.edu/hbase/mincl.html

tryingtolearn1 said:
Homework Statement:: A block starts at rest and slides down a frictionless plane inclined at an angle $\theta$. What should $\theta$ be so that the block travels a given horizontal distance in the minimum amount of time?
Relevant Equations:: $$F=ma$$

When drawing a diagram of the forces acting on the block, I have the following forces: $$\sum F_x =mg\sin\theta = ma$$ $$g\sin\theta = a$$ however the solution has $$F_x = ax = (g \sin\theta) \cos \theta$$ but I am not sure how they got that? I know the normal force is $$N=mg\cos\theta$$ but the normal acts on the y-axis.
gsin##\theta## is the acceleration along the incline.
then what will be the acceleration of the block in the horizontal direction(parallel to the base of the triangle formed by the inclined plane)?

Hamiltonian299792458 said:
gsin##\theta## is the acceleration along the incline.
then what will be the acceleration of the block in the horizontal direction(parallel to the base of the triangle formed by the inclined plane)?

Oh, hmm I kind of understand now but what does it mean to travel a horizontal direction when the plane is inclined?

tryingtolearn1 said:
Oh, hmm I kind of understand now but what does it mean to travel a horizontal direction when the plane is inclined?
just imagine the motion of the block in the x-direction irrespective of its motion in the y .

tryingtolearn1 said:
Oh, hmm I kind of understand now but what does it mean to travel a horizontal direction when the plane is inclined?
JUst as you can resolve a force into horizontal and vertical components, you can do the same with accelerations, velocities and displacements.

haruspex said:
JUst as you can resolve a force into horizontal and vertical components, you can do the same with accelerations, velocities and displacements.

Yes, that I understand but what does it mean physically for the block to travel a horizontal direction? The block is inclined so the only direction it can physically travel is in the direction of the incline so why is the question asking to find the horizontal direction acceleration when the incline acceleration provides a better approximation?

tryingtolearn1 said:
Yes, that I understand but what does it mean physically for the block to travel a horizontal direction? The block is inclined so the only direction it can physically travel is in the direction of the incline so why is the question asking to find the horizontal direction acceleration when the incline acceleration provides a better approximation?
because the question is only concerned with the distance traveled by the block in the horizontal direction(which is not along the incline).

tryingtolearn1 said:
Yes, that I understand but what does it mean physically for the block to travel a horizontal direction? The block is inclined so the only direction it can physically travel is in the direction of the incline so why is the question asking to find the horizontal direction acceleration when the incline acceleration provides a better approximation?
If you move distance s up a slope at θ to the horizontal then you have moved s cos(θ) in the horizontal direction and s sin(θ) in the vertical direction. Your total movement is the sum of the two.

Hamiltonian299792458 said:
because the question is only concerned with the distance traveled by the block in the horizontal direction(which is not along the incline).

Oh I see. I will take the question as is then.

## 1. What is a block slide down a plane?

A block slide down a plane is a basic physics problem that involves a block placed on an inclined plane and the forces acting on the block as it slides down the plane.

## 2. What are the forces acting on a block as it slides down a plane?

The forces acting on a block as it slides down a plane include the force of gravity pulling the block down the plane, the normal force exerted by the plane on the block, and the force of friction between the block and the plane.

## 3. How does the angle of the plane affect the block's motion?

The angle of the plane affects the block's motion by changing the component of the force of gravity acting down the plane. As the angle increases, the force of gravity acting down the plane increases, causing the block to accelerate faster down the plane.

## 4. What is the relationship between the mass of the block and its acceleration down the plane?

The relationship between the mass of the block and its acceleration down the plane is inversely proportional. This means that as the mass of the block increases, the acceleration decreases, and vice versa.

## 5. How can the coefficient of friction affect the block's motion down the plane?

The coefficient of friction can affect the block's motion down the plane by either increasing or decreasing the force of friction between the block and the plane. A higher coefficient of friction will result in a greater force of friction, slowing down the block's motion. On the other hand, a lower coefficient of friction will result in a smaller force of friction, allowing the block to slide down the plane faster.

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