## Student handout: Box Sliding Down Frictionless Wedge

Theoretical Mechanics 2020
Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance $d$ down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.
What students learn
• Lagrangian mechanics will work with non-orthonormal coordinate systems
• To use the dot product to find speed squared for the kinetic energy
• To make sense of their answer by checking dimensions, functional behavior, and special cases

Block Sliding Down a Frictionless Wedge

(Taylor Example 7.5) Consider a block with mass $m$ sliding frictionlessly down an wedge with mass $M$ that makes an angle $\alpha$. The wedge itself slides frictionlessly across a horizontal floor near the surface of Earth. The block is released from the top of the wedge, with both objects initially at rest.

If length of the sloping face is $d$, how long does the block take to reach the bottom?

Keywords
Lagrangian Mechanics Generalized Coordinates Special Cases
Learning Outcomes