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Chapter 11: Problem 17

A sculptor and his assistant are carrying a wedge-shaped marble slab up a flight of stairs, as shown in the flight of stairs, as shown in the figure. The density of the marble is uniform. Both are lifting straight up as they hold the slab completely stationary for a moment. Does the sculptor have to exert more force than the assistant to keep the slab stationary? Explain.

Short Answer

Expert verified
Answer: Yes, the sculptor has to exert more force than the assistant. This is because the center of mass is closer to the base of the wedge, which is towards the sculptor's side, resulting in a greater force ratio.

Step by step solution

01

Identify forces on the slab

First, we need to identify the forces acting on the slab. We have the gravitational force acting on the slab and the forces exerted by the sculptor and the assistant.
02

Find the center of mass

Since the marble slab is wedge-shaped with uniform density, we can determine the center of mass of the object, which is the point where the slab's mass is equally distributed. For a wedge-shaped object, the center of mass is closer to the base, at one-third of its height from that base. Let's assume that the base is towards the sculptor's side.
03

Calculate forces exerted by the sculptor and the assistant

The gravitational force acting on the slab acts through its center of mass. Now, consider the slab as a seesaw, with the center of mass as the pivot point. For the slab to be in equilibrium, the torques due to the forces exerted by the sculptor and the assistant should be equal and opposite. Let the distance between the center of mass and the sculptor's lifting point be x, and the distance between the center of mass and the assistant's lifting point be y. Let the force exerted by the sculptor be F_s and the force exerted by the assistant be F_a. Then: F_s * x = F_a * y
04

Analyze the force ratio

From the equation derived in Step 3, it's clear that the ratio of the force exerted by the sculptor to the force exerted by the assistant is equal to the ratio of the distances from the center of mass to their respective lifting points: F_s/F_a = y/x Since the center of mass is closer to the base (sculptor's side), x is smaller than y, meaning the ratio F_s/F_a is greater than 1.
05

Conclude the answer

From the analysis, we can conclude that the sculptor has to exert more force than the assistant to keep the wedge-shaped marble slab stationary while lifting it up a flight of stairs. This is because the center of mass is closer to the base of the wedge, which is towards the sculptor's side.

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