Question

You give a crate of mass m=1.00kg an initial speed of vi=4.00m/s up an incline of θ=15.2∘ . the crate then slides along the incline, reaches a spring of spring constant k=30.9n/m , and compresses the spring by δs=15.0cm before stopping.

a. assuming there is no friction, what is the distance, d, in meters, along the incline that the crate slides?

Answer 1

To determine the distance along the incline that the crate slides, we need to consider the different forces acting on the crate and the work done by these forces. By equating the work done by the force parallel to the incline to the change in potential energy, we can calculate the distance. Plugging in the given values, we can find the distance d.

Step-by-step explanation:

To determine the distance along the incline that the crate slides, we need to consider the different forces acting on the crate. Since there is no friction, the only force acting on the crate is the component of its weight parallel to the incline, given by:

F_parallel = m * g * sin(theta)

where m is the mass of the crate, g is the acceleration due to gravity, and theta is the angle of the incline. The work done by this force is equal to the change in potential energy of the crate-spring system:

Work = delta potential energy

Since the crate stops when the spring is compressed by delta_s, the potential energy changes from 0 to (1/2) * k * delta_s^2. Equating the work done by the force to the change in potential energy, we have:

F_parallel * d = (1/2) * k * delta_s^2

Solving for d, the distance along the incline that the crate slides, we get:

d = [(1/2) * k * delta_s^2] / (m * g * sin(theta))

Plugging in the given values, we can calculate the value of d.