A box is sliding up an incline that makes an angle of 13.0° with respect to the horizontal. The coefficient of kinetic friction between the box and the surface of the incline is 0.180. The initial speed of the box at the bottom of the incline is 1.80 m/s. How far does the box travel along the incline before coming to rest?
To find out how far the box travels along the incline before coming to rest, we need to apply the laws of motion and the concept of work-energy.
1. Determine the perpendicular and parallel components of the force acting on the box:
The force of gravity acting on the box can be divided into two components: the perpendicular component (mg * cosθ) and the parallel component (mg * sinθ), where m is the mass of the box and θ is the angle of the incline.
2. Calculate the normal force (N) exerted on the box:
The normal force is equal to the perpendicular component of the force of gravity. N = mg * cosθ.
3. Calculate the frictional force (f) opposing the motion:
The frictional force is equal to the coefficient of kinetic friction (μk) multiplied by the normal force. f = μk * N.
4. Calculate the net force (Fnet) acting on the box:
The net force is equal to the parallel component of the force of gravity minus the frictional force. Fnet = mg * sinθ - f.
5. Calculate the acceleration (a) of the box:
The acceleration is equal to the net force divided by the mass of the box. a = Fnet / m.
6. Calculate the distance (d) traveled by the box:
The distance can be calculated using the equation: d = (vi^2 - vf^2) / (2 * a), where vi is the initial velocity and vf is the final velocity (which is 0 in this case).
Now, let's calculate the distance traveled by the box:
Given:
Angle of incline (θ) = 13.0°
Coefficient of kinetic friction (μk) = 0.180
Initial speed (vi) = 1.80 m/s
Mass of the box (m), acceleration (a), normal force (N), and frictional force (f) can be calculated using the formulas mentioned above.
Finally, plug in the values into the distance formula and solve for d.