Write the kinetic energy of the block in the inertial frame. ##\dot y ## is w.r.t. the ( accelerating) wedge.ChiralSuperfields said:
Thank you for your reply @erobz!erobz said:
No, it is incorrect (even including if you add the missing squares). Note that the x and y directions of motion for the block are not orthogonal.ChiralSuperfields said:
No. The wedge is moving with velocity ##\dot x##, the block is riding on the wedge. ##\dot y## is with respect to the wedge, the wedge is not an inertial frame.ChiralSuperfields said:
Thank you for your reply @Orodruin!Orodruin said:
Pretend you are standing in the ground watching this unfold. What velocity do you measure for the block in the horizontal and vertical directions? Remember ##\dot y ## is being measured with respect to the wedge.ChiralSuperfields said:
It most certainly does. Imagine for example that they were parallel. What would the speed be?ChiralSuperfields said:
The drawing is misleading. It shows two arrows labeled ##\dot x##, one on the wedge and another to the right of the sliding block. The first arrow clearly represents the horizontal velocity of the wedge relative to the ground. The second arrow is meant to be the horizontal velocity of the sliding block relative to the ground. It cannot be given the same symbol because it is the horizontal velocity of the sliding block relative to the wedge plus the horizontal velocity of the wedge relative to the ground ##\dot x##. The highlighted term is the cross term that arises when you square the correct expression for the horizontal velocity of the block relative to the ground.ChiralSuperfields said:
erobz said:
Orodruin said:
Thank you for your replies @erobz , @Orodruin and @kuruman!kuruman said:
I doesn’t sound to me like you understand the crux of the problem yet.ChiralSuperfields said:
The cosine rule is perfectly fine to use for addition of velocities here.erobz said:
Ok, I wasn’t familiar with it by name.Orodruin said:
The Lagrange equation is a fundamental equation in classical mechanics that describes the motion of a system in terms of its kinetic and potential energy. It is derived from the principle of least action and is used to derive the equations of motion for a system by expressing them in a form that accounts for constraints and generalized coordinates.
To set up the Lagrange equation for a block on an incline, you first identify the generalized coordinates. For a block sliding down an incline, you can use the angle of the incline (θ) or the position of the block along the incline (s). Next, you determine the kinetic energy (T) and potential energy (V) of the system. The Lagrangian (L) is then defined as L = T - V, and the Lagrange equation is formulated as d/dt(∂L/∂q̇) - ∂L/∂q = 0, where q represents the generalized coordinates.
The kinetic energy (T) of a block sliding down an incline can be expressed as T = (1/2)mv², where m is the mass of the block and v is its velocity. For a block on an incline, the velocity can be related to the angle of the incline and the gravitational acceleration. The potential energy (V) is given by V = mgh, where h is the height of the block above a reference level, which can be expressed in terms of the angle of the incline and the distance along the incline.
Constraints are conditions that limit the motion of a system. In the case of a block on an incline, the constraint is that the block can only move along the surface of the incline. This constraint can be incorporated into the Lagrangian formulation by choosing appropriate generalized coordinates that account for the motion along the incline, which simplifies the analysis and allows for the derivation of the equations of motion without explicitly solving for forces.
To find the equations of motion for a block on an incline using the Lagrange equation, you first calculate the Lagrangian (L = T - V) based on the identified kinetic and potential energies. Then, you apply the Lagrange equation d/dt(∂L/∂q̇) - ∂L/∂q = 0 for the chosen generalized coordinates. This process results in a