- #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.

$$

\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.