consider a cart being pulled up an inclined plane by a student during a Physics lab. The applied force on the cart is 18 N is directed parallel to the incline to cause the cart to be displaced parallel to the incline for a given displacement of 0.7 m. The initial energy plus the work done by the external force equals the final energy. If the cart begins with 0 Joules of energy, and the student does 12.6 Joules of work (F•d•cosine of angle = 18 N•0.7 m•cosine 0 degrees = 12.6 J), then the cart will finish with 12.6 Joules of mechanical energy. The final energy (12.6 J) is equal to the initial energy (0 J) plus the work done by external forces (12.6 J).

I am just wondering, how would the situation differ if there had been friction acting in the opposite direction on the cart?
would the mechanical energy possessed by the cart be less?

If there were friction acting in the opposite direction on the cart, the situation would indeed differ from the previous scenario. Friction opposes the motion and causes a loss of energy in the system.

In the absence of friction, the work done by the external force (18 N) would result in a gain of mechanical energy for the cart. The work done by the force is equal to the force multiplied by the displacement parallel to the incline, multiplied by the cosine of the angle between the force and the displacement.

However, when friction is present, it acts in the opposite direction of motion, which means it does negative work on the cart. The work done by friction is equal to the force of friction multiplied by the displacement parallel to the incline, multiplied by the cosine of the angle between the force of friction and the displacement.

Since the direction of the force of friction is opposite to the direction of motion, the angle between them is 180 degrees, and the cosine of 180 degrees is -1. Therefore, the work done by friction is negative, resulting in a loss of mechanical energy.

If we consider the work done by the student in the presence of friction, the total work done on the cart would be the work done by the external force minus the work done by friction. So, the final energy of the cart would be less than the initial energy because some of the energy is lost to friction.

To summarize, if there is friction acting in the opposite direction on the cart, the mechanical energy possessed by the cart would be less than in the absence of friction. Friction causes a loss of energy in the system, resulting in a decrease in the final mechanical energy of the cart.