Question

1. draw a free body diagram depicting the various forces at play in these static systems: a. mass & pulley system on a frictionless inclined plane: b. gondola lift: c. mass on an inclined plane (with friction):

Explanation:

Step1: Analyze mass - pulley system on frictionless inclined plane

• The mass on the inclined plane has gravitational force $mg$ acting down - ward. This can be resolved into two components: $mg\sin\theta$ along the inclined plane and $mg\cos\theta$ perpendicular to the inclined plane. The normal force $N$ acts perpendicular to the plane, opposing $mg\cos\theta$. The tension $T$ in the string acts along the string, pulling the mass up the plane.

Step2: Analyze gondola lift

• The gondola has a gravitational force $mg$ acting downward. The two segments of the cable exert tension forces $T_1$ and $T_2$ at angles $\theta_1$ and $\theta_2$ respectively with the horizontal.

Step3: Analyze mass on inclined plane with friction

• The mass has gravitational force $mg$ acting downward. Its components are $mg\sin\theta$ along the inclined plane and $mg\cos\theta$ perpendicular to the inclined plane. The normal force $N$ opposes $mg\cos\theta$. The frictional force $f$ acts along the plane, opposing the tendency of the mass to slide. If the mass is on the verge of sliding down, $f$ acts up - the plane; if on the verge of sliding up, $f$ acts down - the plane.

Answer:

a. For the mass - pulley system on a frictionless inclined plane:

• Draw a block on the inclined plane. Draw a vector $mg$ vertically downward. Resolve it into two vectors: $mg\sin\theta$ along the inclined plane and $mg\cos\theta$ perpendicular to the inclined plane. Draw a normal force vector $N$ perpendicular to the plane (opposite to $mg\cos\theta$) and a tension vector $T$ along the string pulling the block up the plane.

b. For the gondola lift:

• Draw the gondola. Draw a vector $mg$ vertically downward. Draw two tension vectors $T_1$ and $T_2$ at angles $\theta_1$ and $\theta_2$ respectively with the horizontal, attached to the gondola.

c. For the mass on an inclined plane with friction:

• Draw a block on the inclined plane. Draw a vector $mg$ vertically downward. Resolve it into $mg\sin\theta$ along the inclined plane and $mg\cos\theta$ perpendicular to the inclined plane. Draw a normal force vector $N$ perpendicular to the plane (opposite to $mg\cos\theta$). Draw a frictional force vector $f$ along the plane, its direction depending on the motion tendency of the block.