Now let’s tackle a couple of questions:

    Adding these two equations together, we find that . Solving for a, we get:

    Because , the acceleration is negative, which, as we defined it, is down for mass M and uphill for mass m.

 

    2. What is the velocity of mass m after mass M has fallen a distance h?

 

    Once again, the in-clined plane is frictionless, so we are dealing with a closed system and we can apply the law of conservation of mechanical energy. Since the masses are initially at rest, . Since mass M falls a distance h, its potential energy changes by –-Mgh. If mass M falls a distance h, then mass m must slide the same distance up the slope of the inclined plane, or a vertical distance of . Therefore, mass m’s potential energy increases by . Because the sum of potential energy and kinetic energy cannot change, we know that the kinetic energy of the two masses increases precisely to the extent that their potential energy decreases. We have all we need to scribble out some equations and solve for v:

    Finally, note that the velocity of mass m is in the uphill direction.

 

    As with the complex equations we encountered with pulley systems above, you needn’t trouble yourself with memorizing a formula like this. If you understand the principles at work in this problem and would feel somewhat comfortable deriving this formula, you know more than SAT II Physics will likely ask of you.

 

 

    There are two significant differences between frictionless inclined plane problems and inclined plane problems where friction is a factor:

1. There’s an extra force to deal with. The force of friction will oppose the downhill component of the gravitational force.
2. We can no longer rely on the law of conservation of mechanical energy. Because energy is being lost through the friction between the mass and the inclined plane, we are no longer dealing with a closed system. Mechanical energy is not conserved.

    Consider the 10 kg box we encountered in our example of a frictionless inclined plane. This time, though, the inclined plane has a coefficient of kinetic friction of . How will this additional factor affect us? Let’s follow three familiar steps:

1. Ask yourself how the system will move: If the force of gravity is strong enough to overcome the force of friction, the box will accelerate down the plane. However, because there is a force acting against the box’s descent, we should expect it to slide with a lesser velocity than it did in the example of the frictionless plane.
2. Choose a coordinate system: There’s no reason not to hold onto the co-ordinate system we used before: the positive x direction is down the slope, and the positive y direction is upward, perpendicular to the slope.
3. Draw free-body diagrams: The free-body diagram will be identical to the one we drew in the example of the frictionless plane, except we will have a vector for the force of friction in the negative x direction.