CALC The potential energy for a particle that can move along the x-axis is U = Ax² + B sin(πx/L), where A, B, and L are constants. What is the force on the particle at (a) x = 0, (b) x = L/2, and (c) x = L?

In a physics lab experiment, a compressed spring launches a 20 g metal ball at a 30° angle. Compressing the spring 20 cm causes the ball to hit the floor 1.5 m below the point at which it leaves the spring after traveling 5.0 m horizontally. What is the spring constant?

CALC An object moving in the xy-plane is subjected to the force $\vec{F}=(2xy\hat{i}+x^2\hat{j})\text{N}$, where x and y are in m. Is this a conservative force?

A particle that can move along the x-axis is part of a system with potential energy U(x) = A/x² − B/x where A and B are positive constants. Where are the particle's equilibrium positions?

CALC An object moving in the xy-plane is subjected to the force $\vec{F}=(2xy\hat{i}+x^2\hat{j})\text{N}$, where x and y are in m. The particle moves from the origin to the point with coordinates (a, b) by moving first along the x-axis to (a, 0), then parallel to the y-axis. How much work does the force do?

A 100 g particle experiences the one-dimensional, conservative force F_x shown in FIGURE P10.60. Suppose the particle is shot to the right from x = 1.0 m with a speed of 25 m/s. Where is its turning point?

A clever engineer designs a 'sprong' that obeys the force law F_x=−q(x−x_eq)³ , where x_eq is the equilibrium position of the end of the sprong and q is the sprong constant. For simplicity, we'll let x_eq = 0 m .Then F_x = −qx³. Find an expression for the potential energy of a stretched or compressed sprong.