Question

A box of mass m is pushed against a spring of negligible mass and force constant k, compressing it a distance x. The box is then released and travels up a ramp that is at an angle α above the horizontal. The coefficient of kinetic friction between the box and the ramp is μk, where μk<1. The box is still moving up the ramp after traveling a distance s>|x| along the ramp.

Calculate the angle α for which the speed of the box after traveling distance s is a minimum.

Answer 1

To find the angle at which the box's speed is minimized, set up an energy balance equation and differentiate with respect to the incline angle, considering Hooke's law, kinetic energy, potential energy, and the work done by friction.

Step-by-step explanation:

The question involves applying concepts of energy conservation and kinetic friction to find the angle at which the speed of a box moving up an incline after being released from a compressed spring is minimized. The initial energy stored in the spring is equal to the kinetic energy of the box plus the work done against friction and the potential energy gained as the box moves up the ramp.

By setting up an equation that equates the initial spring potential energy to the kinetic energy, potential energy, and work done by friction, and differentiating with respect to the angle α, we can solve for the angle that minimizes the speed. The problem involves physics principles such as Hooke's law for springs, the work-energy theorem, and the formula for calculating work done by friction on an inclined plane.