D'Alembert's principle on inclined plane problem

In summary: I think. I sometimes get confused about where the x component of the displacement vector goes in these problems if it is a plane problem. In 3d it's easy. In summary, the virtual displacement for a particle with mass ##m## under the influence of gravitational force and sliding down an inclined plane with angle ##\alpha## is given by ##\delta\vec{r} = \begin{pmatrix} \cos(\alpha) \\ \sin(\alpha) \\ \end{pmatrix} \delta s##, where ##\delta s## is a displacement parallel to the plane. Using d'Alembert's principle, the equations of motion are derived to be ##\ddot{x} \
  • #1
PhysicsRock
114
18
Homework Statement
Examine the motion of a particle with mass ##m## under the influence of the gravitational force ##\vec{F}_g = -mg \vec{e}_y## sliding down an inclined plane with angle ##\alpha##. The particle is initially positioned at ##\vec{r}(0) = (D,H)##.
Derive the equations of motions using d'Alemberts principle.
Relevant Equations
d'Alembert's principle ## \left( m \ddot{\vec{r}} - \vec{F}_i \right) \delta \vec{r} = 0 ##
The virtual displacement should be given by

$$
\delta\vec{r} = \begin{pmatrix} \cos(\alpha) \\ \sin(\alpha) \\ \end{pmatrix} \delta s
$$

where ##\delta s## is a displacement parallel to the plane. The relevant force should be the gravitational force, as given above. Thus, the equations of motion are ought to be

$$
\left[ \begin{pmatrix} m \ddot{x} \\ m \ddot{y} \end{pmatrix} - \begin{pmatrix} 0 \\ -mg \\ \end{pmatrix} \right] \begin{pmatrix} \cos(\alpha) \\ \sin(\alpha) \\ \end{pmatrix} \delta s = m \begin{pmatrix} \ddot{x} \\ \ddot{y} + g \end{pmatrix} \begin{pmatrix} \cos(\alpha) \\ \sin(\alpha) \\ \end{pmatrix} = 0
$$

Doing the multiplications I get

$$
\ddot{x} \cos(\alpha) + (\ddot{y} + g) \sin(\alpha) = 0
$$

Separating that I obtain

$$
\ddot{y} = -g \, , \, \ddot{x} = 0
$$

This, however, describes a free fall with no horizontal acceleration. I must've done something wrong obviously, I just cannot figure out what that is.

Any type of help is highly appreciated. Thank you in advance.
 
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  • #2
PhysicsRock said:
Homework Statement: Examine the motion of a particle with mass ##m## under the influence of the gravitational force ##\vec{F}_g = -mg \vec{e}_y## sliding down an inclined plane with angle ##\alpha##. The particle is initially positioned at ##\vec{r}(0) = (D,H)##.
Derive the equations of motions using d'Alemberts principle.
Relevant Equations: d'Alembert's principle ## \left( m \ddot{\vec{r}} - \vec{F}_i \right) \delta \vec{r} = 0 ##

The virtual displacement should be given by

$$
\delta\vec{r} = \begin{pmatrix} \cos(\alpha) \\ \sin(\alpha) \\ \end{pmatrix} \delta s
$$

where ##\delta s## is a displacement parallel to the plane. The relevant force should be the gravitational force, as given above. Thus, the equations of motion are ought to be

$$
\left[ \begin{pmatrix} m \ddot{x} \\ m \ddot{y} \end{pmatrix} - \begin{pmatrix} 0 \\ -mg \\ \end{pmatrix} \right] \begin{pmatrix} \cos(\alpha) \\ \sin(\alpha) \\ \end{pmatrix} \delta s = m \begin{pmatrix} \ddot{x} \\ \ddot{y} + g \end{pmatrix} \begin{pmatrix} \cos(\alpha) \\ \sin(\alpha) \\ \end{pmatrix} = 0
$$

Doing the multiplications I get

$$
\ddot{x} \cos(\alpha) + (\ddot{y} + g) \sin(\alpha) = 0
$$

Separating that I obtain

$$
\ddot{y} = -g \, , \, \ddot{x} = 0
$$

This, however, describes a free fall with no horizontal acceleration. I must've done something wrong obviously, I just cannot figure out what that is.

Any type of help is highly appreciated. Thank you in advance.
I think you are doing your matrix multiplication incorrectly.

Never mind.
 
Last edited:
  • #3
erobz said:
I think you are doing your matrix multiplication incorrectly.
It's just basic vector multiplication, isn't it? Multiply componentwise and add them up.
 
  • #4
PhysicsRock said:
It's just basic vector multiplication, isn't it? Multiply componentwise and add them up.
Sorry, my bad.
 
  • #5
PhysicsRock said:
The virtual displacement should be given by
Should it? Surely ##\vec r## is three dimensional, and there is a choice of directions to move within the plane.
 
  • #6
haruspex said:
Should it? Surely ##\vec r## is three dimensional, and there is a choice of directions to move within the plane.
Excuse me, I should've mentioned that above, but together with the text comes a schematic that shows that this problem is 2-dimensional, i.e. the displacement only has to account for displacement in the ##x## and ##y## direction.
 
  • #7
I don't know why you set each term equal to zero. I believe you need to use the constraint that:

##\ddot y = \ddot x \tan \alpha##

sub that in and see what you get for ##\ddot x##?

Also, Wiki has the principle quoted as the negative of what you have used.

https://en.wikipedia.org/wiki/D'Alembert's_principle#Special_case_with_constant_mass

I don't know if that matters fundamentally, but it might matter for making sense of the constraints.
 
  • #8
erobz said:
I don't know why you set each term equal to zero. I believe you need to use the constraint that:

##\ddot y = \ddot x \tan \alpha##

sub that in and see what you get for ##\ddot x##?

Also, Wiki has the principle quoted as the negative of what you have used.

https://en.wikipedia.org/wiki/D'Alembert's_principle#Special_case_with_constant_mass

I don't know if that matters fundamentally, but it might matter for making sense of the constraints.
What you said worked. I had the constraint written down, just in the form that it's usually written in, i.e. ##g(\vec{r})=0## and for some reason I just didn't see that I could plug in ##\ddot{x}## in terms of ##\ddot{y}## or vise versa. Thank you a lot.
 
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Likes Lnewqban and erobz
  • #9
PhysicsRock said:
Excuse me, I should've mentioned that above, but together with the text comes a schematic that shows that this problem is 2-dimensional, i.e. the displacement only has to account for displacement in the ##x## and ##y## direction.
ok, so it should have said sliding along a line, not in a plane.
Fwiw, in a plane (standard x, y, z coordinates, x axis lying in the plane) you would have ##\vec{\delta r}=\delta s(\cos(\theta), \sin(\theta)\cos(\alpha), \sin(\theta)\sin(\alpha))##, ##\ddot z=\ddot y\tan(\alpha)##, etc.
 

1. What is D'Alembert's principle on inclined plane problem?

D'Alembert's principle on inclined plane problem is a principle in physics that states that the net force acting on an object on an inclined plane is equal to the product of the mass of the object and its acceleration. This principle is based on the laws of motion and is often used to analyze the motion of objects on inclined planes.

2. How is D'Alembert's principle used to solve problems on inclined planes?

D'Alembert's principle is used to simplify the analysis of motion on inclined planes by considering the forces acting on the object as if they were acting on a horizontal plane. This allows for the use of basic equations of motion, such as Newton's second law, to solve for the acceleration and other quantities of interest.

3. What are the assumptions made when applying D'Alembert's principle on inclined planes?

The main assumptions made when using D'Alembert's principle on inclined planes are that the object is in equilibrium or moving at a constant velocity, and that there is no friction or other external forces acting on the object. These assumptions allow for a simplified analysis of the forces acting on the object.

4. Can D'Alembert's principle be applied to objects on inclined planes with friction?

Yes, D'Alembert's principle can still be applied to objects on inclined planes with friction. However, the frictional force must be taken into account as an external force acting on the object. This may require additional equations or calculations to determine the acceleration and other quantities of interest.

5. What are some real-life examples of D'Alembert's principle on inclined planes?

D'Alembert's principle on inclined planes can be observed in many real-life situations, such as a car driving up a hill, a rollercoaster moving up and down inclines, or a person skiing down a slope. In all of these examples, the forces acting on the object can be analyzed using D'Alembert's principle to determine the acceleration and other factors affecting the motion.

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