Question

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

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

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.

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.

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

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.

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.

Key concepts

Gravitational Force

Gravitational force is a fundamental concept in physics. It is the force of attraction between two masses. For objects on Earth, this force is primarily the pull towards the center of the planet. It can be calculated using the formula:

• Force of gravity, \( F = m \, g \)
• where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity on Earth \( (9.8 \, \text{m/s}^2) \)

In the context of the wedge-shaped marble slab, the gravitational force acts uniformly across the slab's mass and is centered at the slab's center of mass. Understanding how gravitational force interacts with objects in different shapes, like the wedge, helps us analyze equilibrium conditions. As the sculptor and assistant lift the slab, they're effectively counterbalancing this force. This ensures the slab remains stationary as they carry it up the stairs.

Torque and Equilibrium

Torque is a measure of the rotational force acting on an object. It is calculated as the product of force and the distance from the pivot point, also known as the moment arm. In mathematical terms:

• Torque, \( \tau = F \, d \), where \( F \) is the force applied and \( d \) is the perpendicular distance from the pivot

To maintain equilibrium, an object's net torque must be zero, meaning the torques in opposite directions must balance each other out. In the case of the slab, the center of mass acts as a pivot point, with the sculptor and assistant's forces effectively acting like a seesaw. The equation:

• \( F_s \times x = F_a \times y \)

shows that to keep the slab in equilibrium, the product of the force and distance from the pivot must be equal on both sides. Since the center of mass is closer to the sculptor (x < y), the sculptor must exert a greater force than the assistant to maintain balance. This ensures the slab remains stationary and doesn't rotate as they lift it.

Wedge-Shaped Objects

Wedge-shaped objects have unique properties due to their geometry. The shape affects how forces and torques act and influence the object's stability. For wedge-shaped objects with uniform density, the center of mass is typically located closer to the base. This is a vital point for analyzing equilibrium and balance. When considering the slab, knowing the position of the center of mass helps determine how forces interact with it. Because it's closer to the sculptor's side, it impacts how much force each person must exert to keep the object stable. Understanding the shape's influence on the center of mass allows one to apply principles of torque and gravitational force effectively. When dealing with such shapes, always examine:

• The position of the center of mass relative to the base
• How this affects the distribution of forces in scenarios involving lifting or rotation

These aspects help in solving real-world problems involving transportation and stabilization of irregular objects like wedge-shaped slabs.