A mass,m,slides down a frictionless incline and hits a spring with spring constant k.Find the spring compression when the acceleration of the mass is zero.

Question

Quizplus · Accepted Answer

When the mass slides down the incline and hits the spring, its potential energy is converted to kinetic energy. The kinetic energy of the mass is given by K = (1/2)mv^2, where m is the mass of the object and v is its velocity. Since the incline is frictionless, there is no loss of energy due to friction. The kinetic energy of the mass is transferred to the spring, which compresses.

The amount of compression of the spring can be calculated by equating the potential energy stored in the spring to the kinetic energy of the mass at the point where the acceleration of the mass becomes zero. At this point, all the kinetic energy of the mass is converted to potential energy stored in the spring.

The potential energy stored in a spring of stiffness k and compressed by a distance x is given by U = (1/2)kx^2.

Equating the initial kinetic energy of the mass to the potential energy stored in the spring when the acceleration of the mass becomes zero, we get:

(1/2)mv^2 = (1/2)kx^2

Since the acceleration of the mass is zero, the force on the mass due to the spring is equal and opposite to the gravitational force on the mass. This gives:

mg sinθ = kx

Solving for x, we get:

x = (mg sinθ) / k

Thus, the spring compression is given by x = (mg sinθ) / k. This is the same as option D.

A Mass,m,slides Down a Frictionless Incline and Hits a Spring θ\thetaθ

A Mass,m,slides Down a Frictionless Incline and Hits a Spring $\theta$