In this diagram, V represents the force due to friction. The equation for the force due to friction is , where is the coefficient of friction.

In this case, Z represents the normal force. We can re-write the equation for friction:

Z can be re-written in terms of the angle, but will always need to be multiplied by the coefficient of friction in order to give an equation for V.

The other equations are true.

- X and Y form a right angle, so the Pythagorean theorem applies.

- Z is the normal force, which is, by definition, equal and opposite the vertical force of gravity.

- W is the total force of gravity, which will be equal to the mass times the acceleration of gravity.

- the triangle formed by W, X, and Y is similar to the triangle formed by the surface, Q, and P, meaning that these angles must be equal.

In this problem, Y and Z are equal, but opposite forces, and Z is our normal force.

If we can solve for Y, then we can find Z.

We can use our understanding of trigonometry to find an equation for Y.

If we plug in  for the angle, we see:

Since we are solving for Y, we can multiply both sides by W.

Now that we know an equation for Y, we can return to our original equation to solve for Z.

There is no way for us to know the distance the object travels. Even though the force diagram places our object roughly halfway up the plane, we do not know the position where the object starts or where the object ends. We cannot calculate a change in distance.

We can use our understanding of trigonometry to find this relationship, since X and Y form a right angle.

If we plug in  for the angle, we see that X is opposite of and W is the hypotenuse.

Since we're looking for X, we can multiply both sides by W.

For this problem, we need to break the forces into horizontal and vertical components. The only difference with an inclined plane is that you have to translate the horizontal plane to be parallel to the surface upon which the object is traveling, and the vertical plane to be perpendicular to the surface upon which the object is traveling.

Luckily, this is already done in the diagram, with the force of gravity (W) broken into the vertical component (Y) and horizontal component (X).

Now, we need to sum the forces in each plane. The forces in the vertical plane will be perpendicular to the surface: Z and Y.

Since Z and Y are opposite and equal forces:

There is no net force in the vertical plane. This makes sense, as we would not expect the object to have any movement perpendicular to the surface on which it is sliding.

Now we can sum the horizontal forces: V and X.

Note that X is positive and V is negative. This is because V represents the force of friction, and is opposite to the force of X.

Since there is no force in the vertical plane, this gives our final net force: .

The zeroth law of thermodynamics states that if two objects are in thermodynamic equilibrium with a third object, then they must be in thermodynamic equilibrium with each other. In this question, the coffee is in equilibrium with both the milk and the sugar, allowing us to conclude that the milk and sugar must be in equilibrium with each other.

The second law of thermodynamics states that the entropy of the universe is always increasing, and is not relevant to this particular scenario.

The zeroth law of thermodynamics states that if two separate systems are in equilibrium with a third system, then they are in equilibrium with each other. The zeroth law of thermodynamics is essentially equivalent to the transitive property of mathematics.

If and , then .

The first law of thermodynamics states that internal energy changes due to heat flow. Mathematically, this law is presented as .

The second law of thermodynamics states that the entropy (or disorder) of the universe is always increasing. Certain systems exist in which there is a local decrease in entropy, but these processes are always balanced by an increase of entropy outside of the system.

The third law of thermodynamics states that absolute zero is the state in which a system has zero entropy. Essentially, this means that it is impossible to reach absolute zero (at least with modern technology).

The zeroth law of thermodynamics states that if two systems are in thermal equilibrium with a third system, they are in equilibrium with each other. If gas A is in equilibrium with gas B and gas C, then gas be and gas C must be in thermal equilibrium with each other.

For this problem, use the first law of thermodynamics. The change in energy equals the increase in heat energy minus the work done.

We are not given a value for work, but we can solve for it using the force and distance. Work is the product of force and displacement.

Now that we have the value of work done and the value for heat added, we can solve for the total change in energy.

For this problem, use the first law of thermodynamics. The change in energy equals the increase in heat energy minus the work done.

We are given the amount of work done by the gas and the total energy of the system. Using these values, we can solve for the heat added.