Question

A sculpture is \(4.00 \mathrm{m}\) tall and has its center of gravity located \(1.80 \mathrm{m}\) above the center of its base. The base is a square with a side of \(1.10 \mathrm{m} .\) To what angle \(\theta\) can the sculpture be tipped before it falls over? (W) tutorial: filing cabinet)

Short answer

The sculpture can be tipped to approximately 16.98° before it falls over.

Step-by-step solution

Visualize the Problem

Imagine the sculpture as a tall rectangle standing on a square base. When tilted, it pivots around one of its base edges. The tipping point or the angle at which it tips over is when the center of gravity is vertically over the base's edge.

Establish Coordinates and Geometry

Since the base is a square with side length, we have half the base as \( \frac{1.10}{2} = 0.55 \, \mathrm{m} \). The center of gravity is 1.8 m above the center of the base. In a right triangle formed by tipping, the horizontal distance from the center of gravity to the base edge becomes essential as it approaches this distance.

Define the Right Triangle

In a right triangle formed when the sculpture is about to tip, we have:- The height of the triangle from the center of gravity to the pivot point is \(1.8 \, \mathrm{m}\).- The base of this triangle is \(0.55 \, \mathrm{m}\).The tangent of the angle \(\theta\) is given by: \[\tan (\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{0.55}{1.8}\]

Calculate the Angle

To find \ \theta\, take the inverse tangent (arctan) of the ratio:\[\theta = \tan^{-1} \left(\frac{0.55}{1.8}\right)\]Use a calculator to compute the angle \ \theta\.

Key concepts

Center of Gravity

The center of gravity is a critical concept in physics that describes the point in an object where all its weight is concentrated. It acts as if all the mass of the object is centered there. In practical terms, it helps us predict how an object will behave under the influence of gravity, especially when balancing or rotating.

For the sculpture in our problem, the center of gravity is located 1.8 meters above the base. This location determines how the object will balance and at what point it might tip over when tilted. When you think about tipping, picture how the gravitational force pulls downwards through this point. If it moves outside the base, the sculpture will tip.

• All forces and weight are assumed to act through the center of gravity.
• In stable conditions, the center of gravity lies above the base of support.

This concept simplifies complex systems and is crucial in engineering, architecture, and daily life scenarios like carrying a tray without spilling.

Statics and Equilibrium

Statics is the study of forces in a system at rest. Equilibrium occurs when all forces in a system are balanced, resulting in no movement. In our example, the sculpture remains stationary as long as the forces are balanced around its tipping point.

In the context of the sculpture, equilibrium means the gravitational center stays above the base. When tipped, it maintains this position until a critical point. At this stage, a small change will cause it to tip. The key elements of equilibrium include:

• Static Equilibrium: Objects are in rest with no net force acting on them.
• Rotational Equilibrium: There is no net torque, meaning the object does not rotate.

Understanding these elements allows us to keep objects stable. In practical applications, it aids in designing structures that do not easily tip or wobble.

Tipping Angle

The tipping angle is the maximum angle at which an object can be tilted before it begins to tip over. It reflects the balance between gravitational forces and the object's base dimensions. When the angle is exceeded, the object becomes unstable and falls.

For the sculpture, we calculate the angle by using the tangent of an angle in the right triangle formed when the sculpture is nearly tipping. The key steps include:

• Determining the theoretical triangle: The vertical side equals the height of the center of gravity, and the horizontal side equals half of the base.
• Finding the tangent of this angle (\( heta\)) using the ratio \[\tan (\theta) = \frac{0.55}{1.8}\]
• Using inverse tangent (arctan) to calculate \(\theta\). This is the tipping angle.

For practical safety measures in design and construction, understanding and calculating the tipping angle can prevent accidents by ensuring that objects remain stable up to a certain degree of tilt.

Geometry in Physics

Geometry plays a vital role in understanding physics problems by offering a framework to visualize forces and movements in space. The tipping problem is a geometric puzzle at heart, with shapes, lines, and angles helping to resolve the forces acting on objects.

In our sculpture scenario, we use:

• A square base to visualize stability regions.
• A right triangle to understand the tipping mechanism.

The right triangle formed during tipping contains:

• The vertical leg (1.8 m) representing the height of the center of gravity.
• The horizontal leg (0.55 m) which is half of the base's side.

Geometry helps in achieving accurate calculations necessary to solve physical problems. By applying geometric principles, we break down complex systems into manageable parts, thereby understanding movement, forces, and balance.