State the work-energy theorem in words and apply it to calculate the kinetic energy of a crate. Also, draw a free-body diagram for the crate on an incline.

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.

The crate starts at rest at point A, so its initial kinetic energy $K_A = 0 \text{ J}$.
A constant horizontal force $F = 12 \text{ N}$ acts on the crate over a distance $d_{AB} = 7 \text{ m}$.
The surface from A to B is frictionless.
The work done by the force $F$ is given by:

According to the work-energy theorem, the net work done on the crate is equal to the change in its kinetic energy:

Since $K_A = 0 \text{ J}$ and the only force doing work is $F$:

The kinetic energy of the crate at point B is $\boxed{\text{84 J}}$.

As the crate moves down the incline, the forces acting on it are:

• Gravitational force (Weight): Acts vertically downwards from the center of mass of the crate.
• Normal force: Acts perpendicular to the inclined surface, pointing away from the surface.
• Kinetic frictional force: Acts parallel to the inclined surface, pointing upwards along the incline (opposite to the direction of motion).

From point B to C, the crate moves down an incline.
Initial kinetic energy at B: $K_B = 84 \text{ J}$
Mass of the crate: $m = 3 \text{ kg}$
Distance from B to C: $d_{BC} = 6.8 \text{ m}$
Angle of inclination: $\theta = 20^\circ$
Constant frictional force: $f_k = 21 \text{ N}$
Acceleration due to gravity: $g = 9.8 \text{ m/s}^2$

We use the work-energy theorem from B to C: $W_{net, BC} = K_C - K_B$.
The forces doing work are gravity and friction.