Analytical Mechanics: Solving Equations of Motion for Block Sliding Down Ramp

In summary: No, in summary, the block is sliding down a long straight ramp and is experiencing frictional forces and air resistance. At the end of the slide, the block reaches a terminal velocity. Using the equations of motion, determine the speed of the block in terms of the variables given.
  • #1
tarletontexan
30
0

Homework Statement


a block(mass m), is sliding down a long straight ramp with angle theta, this block is undergoing frictional forces(mu) and air resistance proportional to the speed of the block. In its sliding the block reaches a terminal velocity. Determine the equation of motion and determine the speed in terms of those given variables


Homework Equations


-bV = air resistance
muFn= frictional force
fnsin(theta)-(frictional force+air resistance)=ma


The Attempt at a Solution


I'm completely stuck...I don't know where to start i have the equations i think are the equations of motion but am not sure. If someone could help me out that would be wonderful
 
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  • #2
You seem to be counting the frictional force twice for some reason and have completely ignored gravity. You should also have an equation for the vertical forces on the block in order to determine the normal force.
 
  • #3
No, i understand the normal force is Fn(normal force) cos(theta). And as for the equations that's just my understanding of the forces on the block, the one set equal to (ma) is the equation for motion of my system as i believe...
 
  • #4
tarletontexan said:
No, i understand the normal force is Fn(normal force) cos(theta).

That expression is actually the gravitational force mg. You seem to be using the term normal force for 2 different things.

And as for the equations that's just my understanding of the forces on the block, the one set equal to (ma) is the equation for motion of my system as i believe...

Your equation would be correct if you replaced fn by mg.

Once you sort the equation out, do you see how to solve for the velocity?
 
  • #5
ok but I'm not seein it? the block is on a ramp, so with the incline mg is straight down while the normal force is perpendicular to the surface of the ramp at angle theta. So i don't quite see how to solve for the velocity...
 
  • #6
tarletontexan said:
ok but I'm not seein it? the block is on a ramp, so with the incline mg is straight down while the normal force is perpendicular to the surface of the ramp at angle theta. So i don't quite see how to solve for the velocity...

Write down the equations for the forces horizontal to the ramp and vertical with respect to the ramp. The vertical equation let's you write the normal force in terms of the gravitational force. Express the horizontal forces in terms of the gravitational force, the coefficient of friction and the air resistance. This gives you an equation relating the acceleration to the velocity.
 
  • #7
ok I see now, so the equations now would be mgcos(theta)= fn, and -mgcos(theta)(uK)+(-bv)+mgsin(theta)=ma
 

1. What is analytical mechanics?

Analytical mechanics is a branch of classical mechanics that uses mathematical equations to describe and predict the motion of systems. It involves solving equations of motion using principles such as Newton's laws of motion and conservation of energy and momentum.

2. How do you solve equations of motion for a block sliding down a ramp?

To solve equations of motion for a block sliding down a ramp, you first need to identify all the forces acting on the block, including gravity, normal force, and friction. Then, you can use Newton's laws of motion to write equations of motion for the block in both the x and y directions. Finally, you can use calculus to solve these equations and determine the block's position, velocity, and acceleration at any given time.

3. What factors affect the block's motion down the ramp?

The block's motion down the ramp is affected by several factors, including the angle of the ramp, the mass of the block, the coefficient of friction between the block and the ramp, and the initial conditions of the block (e.g. initial position and velocity).

4. How does the angle of the ramp impact the block's acceleration?

The angle of the ramp affects the block's acceleration because it determines the component of the force of gravity that is acting to pull the block down the ramp. As the angle of the ramp increases, the force of gravity acting in the direction of motion also increases, resulting in a greater acceleration.

5. Can analytical mechanics be used to solve more complex systems?

Yes, analytical mechanics can be used to solve complex systems involving multiple objects and forces. It can also be extended to include more advanced concepts such as rotational motion and systems with non-conservative forces. However, for extremely complex systems, numerical methods may be more effective for finding solutions.

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