Block on inclined plane acceleration without rotating coordinates

In summary, the block experiences an acceleration in the x and y directions, but the magnitude of the acceleration is incorrect.
  • #1
brainpushups
452
194

Homework Statement


I was doing this for fun, but this seemed like the right place to post. Typically inclined plane problems are solved by rotating the coordinate system so the x-axis is along the plane. I decided to try solving the problem without rotating the coordinates. In this case there should be acceleration in the x and y direction, and the magnitude of the acceleration vector should return the same value as the problem solved with rotating the coordinates. My attempts at solving the problem with "normal" coordinates have been incorrect and I'd appreciate any insight as to what I'm doing wrong.

If I need to state the problem than here it is: A block mass m is on an inclined plane at angle [tex]\Theta[/tex]. Find components of the acceleration of the block without rotating the coordinates and also find the magnitude of the acceleration.

http://dots.physics.orst.edu/graphics/image_maps/inclined_plane.gif [Broken]


Homework Equations


magnitude of the acceleration of a block down a plane using rotated coordinates: a = g Sin[[tex]\Theta[/tex]]
magnitude of the normal force: N = mg Cos[[tex]\Theta[/tex]]


The Attempt at a Solution


The sum of the forces in the x direction only consist of the x component of the normal force on the block
[tex]\Sigma[/tex]Fx = Nx= -mgCos[[tex]\Theta[/tex]]Cos[[tex]\Theta[/tex]]
or Nx=-mgCos2[tex]\Theta[/tex]=max
The extra cos theta exists because the magnitude of the normal force is mgCos[[tex]\Theta[/tex]]
From this, ax= -gCos2[tex]\Theta[/tex]
The sum of the forces in the y direction are the gravitational force and the y component of the normal force
[tex]\Sigma[/tex]Fy= -mg + mgCos[[tex]\Theta[/tex]]Sin[[tex]\Theta[/tex]]
From this, ay= -g+gCos[tex]\Theta[/tex]Sin[tex]\Theta[/tex]
or if you wish to apply the double angle formula:
ay= -g+1/2gSin[2[tex]\Theta[/tex]]


Clearly these acceleration components do not give the correct value for the magnitude of the acceleration. A zero degree angle returns an acceleration in the -x direction equal to g. What did I miss?

PS. I apologize if there are any mistakes or this is hard to read. This is my first post.
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
You messed up the calculation of the x & y components of the normal force. Note that the normal force makes an angle Θ with the vertical, not the horizontal.
 
  • #3
Thanks for the reply. Careless errors like that can be so frustrating!
 

1. What is a block on an inclined plane?

A block on an inclined plane refers to a physical system where a block is placed on a sloped surface. The surface can be at any angle to the horizontal plane, and the block may or may not be in motion.

2. How does the acceleration of the block on an inclined plane change?

The acceleration of the block on an inclined plane changes depending on the angle of the incline and the force acting on the block. If the plane is frictionless, the acceleration will be constant and equal to the component of the force acting parallel to the incline. If there is friction present, the acceleration will decrease as the incline angle increases.

3. What factors affect the acceleration of the block on an inclined plane?

The acceleration of the block on an inclined plane is affected by the angle of the incline, the mass of the block, the force acting on the block, and the presence of friction. These factors can either increase or decrease the acceleration of the block.

4. How is the acceleration of the block on an inclined plane calculated?

The acceleration of the block on an inclined plane can be calculated using Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. In the case of a block on an inclined plane, the net force is equal to the component of the force acting parallel to the incline, and the acceleration is in the same direction as this force.

5. What are some real-life applications of a block on an inclined plane?

A block on an inclined plane has many practical applications, such as ramps and chutes used in construction, conveyor belts in factories, and roller coasters at amusement parks. Understanding the acceleration of a block on an inclined plane is also essential in fields like engineering, physics, and mechanics.

Similar threads

  • Introductory Physics Homework Help
Replies
10
Views
692
  • Introductory Physics Homework Help
Replies
14
Views
794
  • Introductory Physics Homework Help
Replies
12
Views
878
  • Introductory Physics Homework Help
Replies
18
Views
3K
  • Introductory Physics Homework Help
Replies
2
Views
626
  • Introductory Physics Homework Help
2
Replies
44
Views
3K
  • Introductory Physics Homework Help
Replies
4
Views
853
  • Introductory Physics Homework Help
Replies
11
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
929
  • Introductory Physics Homework Help
Replies
4
Views
1K
Back
Top