Problem 71 - Physics

Question

A \(3500-\mathrm{kg}\) statue is placed on top of a cylindrical concrete \(\left(Y=2.3 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}\right)\) stand. The stand has a crosssectional area of \(7.3 \times 10^{-2} \mathrm{~m}^{2}\) and a height of \(1.8 \mathrm{~m}\). By how much does the statue compress the stand?

Step-by-Step Solution

Answer

The stand compresses by approximately \(3.68 \times 10^{-5}\) meters.

Step 1: Identify the given values

We begin by identifying the quantities provided in the problem: The mass of the statue is \( m = 3500 \) kg. The Young's modulus of concrete is \( Y = 2.3 \times 10^{10} \, \mathrm{N/m}^2 \). The cross-sectional area of the stand is \( A = 7.3 \times 10^{-2} \) m\(^2\). The original height of the stand is \( L_0 = 1.8 \) m.

Step 2: Calculate the force exerted by the statue

The force \( F \) exerted by the statue due to gravity is calculated using the formula \( F = mg \), where \( g \) is the acceleration due to gravity (\( 9.81 \) m/s\(^2\)).\[ F = 3500 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 34335 \, \text{N} \]

Step 3: Use Young's Modulus to find compression

The amount by which the statue compresses the stand is given by the formula \( \Delta L = \frac{FL_0}{YA} \), where \( \Delta L \) is the change in length (compression).Substitute the values:\[ \Delta L = \frac{34335 \, \text{N} \times 1.8 \, \text{m}}{2.3 \times 10^{10} \, \text{N/m}^2 \times 7.3 \times 10^{-2} \, \text{m}^2} \]

Step 4: Perform the calculation

Now calculate the change in length:\[ \Delta L = \frac{61803 \, \text{N} \cdot \, \text{m}}{1.679 \times 10^9 \, \text{N}} \approx 3.68 \times 10^{-5} \, \text{m} \]This means the stand compresses by approximately \(3.68 \times 10^{-5} \) meters.

Key Concepts

Force Calculation

The calculation of force is central to many problems in physics, including the one involving the statue and the cylindrical concrete stand. To begin with, the force exerted by an object due to gravity is determined by its mass and the acceleration due to gravity. This force is commonly referred to as the gravitational force. For our specific problem, the formula used is:- \( F = mg \)Here, \( m \) is the mass of the statue, which is 3500 kg, and \( g \) is the acceleration due to gravity, generally approximated as 9.81 m/s\(^2\) on Earth's surface. By substituting these values into the equation, we obtain the force applied by the statue:- \( F = 3500 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 34335 \, \text{N} \)This calculation tells us that the statue exerts a downward force of 34,335 Newtons on the concrete stand due to gravity. Understanding this force is vital for calculating subsequent compressive forces using Young's Modulus.

Compression

Compression involves the shortening of an object under the action of an applied force. In this problem, we are interested in how much the concrete stand compresses due to the force exerted by the statue. The force results in a decrease in height of the stand, known as compression.To calculate the compression, we utilize Young's Modulus, a property that describes how a material deforms under stress. The relationship is given by:- \( \Delta L = \frac{FL_0}{YA} \)Where- \( \Delta L \) is the change in length,- \( F \) is the force applied (34335 N from our previous calculation),- \( L_0 \) is the original length of the stand (1.8 m),- \( Y \) is Young's Modulus for concrete \( (2.3 \times 10^{10} \, \text{N/m}^2) \),- \( A \) is the cross-sectional area of the stand \( (7.3 \times 10^{-2} \) m\(^2\)).Substituting the values, we can find:- \( \Delta L = \frac{34335 \, \text{N} \times 1.8 \, \text{m}}{2.3 \times 10^{10} \, \text{N/m}^2 \times 7.3 \times 10^{-2} \, \text{m}^2} \)This calculation yields the stand's compression, which illustrates how structural elements like a concrete stand respond to heavy loads.

Material Properties

Understanding material properties, such as Young's Modulus, is critical when analyzing how materials behave under different forces. Young's Modulus, \( Y \), is a measure of a material's stiffness, detailing how much a material will deform under stress.In the context of this exercise:- Concrete has a Young's Modulus of \( 2.3 \times 10^{10} \, \text{N/m}^2 \).- This high value suggests that concrete is quite resistant to deformation under force.Material properties give insight into the suitability of a material for specific applications. In structural engineering, concrete is often chosen for its high stiffness and ability to bear substantial loads without significant deformation. However, it is important to determine how much a structure will compress under a load to ensure safety and structural integrity. Young's Modulus provides the quantitative means to calculate expected deformations, ensuring that designs meet necessary criteria for strength and durability.