8.4 Potential Energy Diagrams and Stability

Repeat the example when the particle’s mechanical energy is $+ 0.25 \text{J}.$

Find $x \left(t\right)$ for the mass-spring system in the example if the particle starts from $x_{0} = 0$ at $t = 0 .$ What is the particle’s initial velocity?

A mysterious constant force of 10 N acts horizontally on everything. The direction of the force is found to be always pointed toward a wall in a big hall. Find the potential energy of a particle due to this force when it is at a distance x from the wall, assuming the potential energy at the wall to be zero.

A particle of mass 4.0 kg is constrained to move along the x-axis under a single force $F \left(x\right) = − c x^{3} ,$ where $c = 8.0 \text{N}/\text{m}^{3} .$ The particle’s speed at A, where $x_{A} = 1.0 \text{m},$ is 6.0 m/s. What is its speed at B, where $x_{B} = −2.0 \text{m}?$