A box is sliding up an incline that makes an angle of 15.0 degrees 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 and the surface of the incline is 1.50 m/s. How far does the box travel along the incline before coming to rest?
This is from Chapter 4, Forces and Newton's Laws of Motion.
Should I redefine the plane to make the x-axis the same line as the incline?
What is F(fr)?
Yes, it would be helpful to redefine the coordinate system with the x-axis aligned along the incline. This will simplify the calculations since the force of gravity can be resolved into components parallel and perpendicular to the incline.
In this question, it would be helpful to redefine the coordinate system so that the x-axis is aligned with the incline. By doing this, it will simplify the analysis of the problem.
To redefine the plane and align the x-axis with the incline, you can rotate the coordinate system by an angle equal to the angle of the incline. In this case, the incline makes an angle of 15.0 degrees with respect to the horizontal.
To align the x-axis with the incline, you will want to rotate the coordinate system clockwise by an angle of 15.0 degrees. This means that any vector or position initially in the horizontal direction will now have components in both the x and y directions.
Once you have redefined the coordinate system, you can break down all the forces acting on the box along the incline. These forces include the gravitational force, the normal force, and the kinetic friction.
To find the distance the box travels along the incline before coming to rest, you need to use the concept of work and energy. The work done on the box by the force of kinetic friction is equal to the change in kinetic energy of the box.
You can calculate the work done by kinetic friction using the formula: work = force * distance. In this case, the force of kinetic friction is equal to the coefficient of kinetic friction multiplied by the normal force, and the distance is the distance along the incline.
To find the normal force, you need to resolve the gravitational force into components perpendicular and parallel to the incline. The component of the gravitational force perpendicular to the incline is equal to m * g * cos(theta), where m is the mass of the box, g is the acceleration due to gravity, and theta is the angle of the incline.
Using the relation between force and acceleration along the incline, you can relate the force of kinetic friction to the net force acting on the box. This allows you to solve for the acceleration along the incline.
Once you have the acceleration, you can use the equations of motion to find the time it takes for the box to come to rest. Finally, you can use this time to calculate the distance traveled by the box along the incline using the equation: distance = initial velocity * time + (1/2) * acceleration * time^2.
By following these steps, you can find the distance the box travels along the incline before coming to rest.
N=mgcosα
F(fr)= μN = μmgcosα
ma=F(fr)+mgsinα =>
ma= μmgcosα + mgsinα
a=g(μcosα+sinα)=....
s=v²/2a=....