Question

directions: solve the problem shown below. any data, calculations, or models you use to support your thinking should be explained. problem: using the three methods shown below, a small block is raised from rest on the ground until it is again at rest a distance ( d ) above the ground. in method 1, a rope pulls the block straight up. in methods 2 and 3, the block is pushed up frictionless ramps of different lengths. which of the following correctly ranks the total work ( w ) done on the block by external forces as the block is raised using each method? explain. 1. ( w_1 = w_2 = w_3 ) 2. ( w_1 > w_2 > w_3 ) 3. ( w_1 > (w_2 = w_3) ) 4. ( w_3 > w_2 > w_1 )

Explanation:

To determine the work done on the block, we use the work - energy theorem, which states that the net work done on an object is equal to its change in kinetic energy ($W_{net}=\Delta KE$). The block starts from rest and ends at rest, so $\Delta KE = 0$. This means the net work done on the block in each case is zero.

For each method, the forces acting on the block are the external force and gravity. Let the external force be $F$ and the force of gravity be $mg$ (where $m$ is the mass of the block and $g$ is the acceleration due to gravity). The work done by the external force $W_F$ and the work done by gravity $W_g$ must satisfy $W_F+W_g = 0$ (since $W_{net}=0$), so $W_F=-W_g$.

The work done by gravity is given by $W_g=-mgh$, where $h$ is the vertical displacement. In all three methods, the vertical displacement of the block is $d$ (it is raised to a height $d$ above the ground). So, $W_g=-mgd$ for each method. Then, the work done by the external force $W_F = mgd$ for each method.

In Method 1, the external force is used to lift the block straight up. In Methods 2 and 3, the block is pushed up frictionless ramps. Even though the distance along the ramp (the displacement for the external force) is different in Methods 2 and 3, the vertical displacement (which determines the work done against gravity) is the same ($d$) for all three methods. Since the ramps are frictionless, there is no non - conservative force doing work. So, the work done by the external force in each case is equal to the change in gravitational potential energy, which depends only on the vertical height $d$. Therefore, the work done by the external forces in all three methods is the same.

Answer:

1. $W_1 = W_2 = W_3$