Terminal velocity of heavy and large objects. You drop two spheres from a high tower. First, assume that the spheres have the same diameter, \(d\), and surface properties, so that they have the same air resistance, but they have different masses, \(m_{A}\) and \(m_{B}\). The air resistance is described using a quadratic law with the coefficient \(D\) for both spheres. (a) Draw a free-body diagram for a sphere as it is falling. (b) Find an expression for the acceleration of either sphere. (c) Which object has the largest acceleration-the object with the largest or with the smallest mass? Now, let us modify the experiment. We now drop two spheres of different diameter, \(d_{1}\) and \(d_{2}\), but the spheres are solid and made of the same materials, for example steel. They will therefore have different masses, \(m_{1}\) and \(m_{2}\). Still, air resistance for both spheres are described using a quadratic law, but the coefficient \(D\) depends on cross-sectional area of the sphere, and therefore on the diameter: \(D=C_{0} d^{2}\), where \(C_{0}\) is a constant. (d) Find an expression for the acceleration of such a sphere as a function of the diameter of the sphere. (e) Which object has the largest acceleration-the object with the largest or with the smallest diameter?

Step by step solution

01

- Drawing a Free-body Diagram (Part a)

First, draw a free-body diagram for one of the spheres. Include the force of gravity acting downwards, labeled as mg, and the air resistance force acting upwards, labeled as Dv^2 (since air resistance is quadratic).

02

- Understanding Forces (Part b)

Realize that the net force acting on the sphere is the difference between the gravitational force and the air resistance force. Mathematically, this is given by: F_net = mg - Dv^2.

03

- Expression for Acceleration (Part b)

Using Newton's second law, F = ma, rewrite the net force equation as: ma = mg - Dv^2. Therefore, the acceleration (a) can be found by rearranging this equation: a = g - (Dv^2)/m.

04

- Compare Accelerations (Part c)

To determine which object has the largest acceleration, compare the expression a = g - (Dv^2)/m for both masses. Since the mass m_B is larger than m_A, the term (Dv^2)/m_B is smaller than (Dv^2)/m_A. Therefore, a_B > a_A. Thus, the object with the largest mass has the largest acceleration.

05

- Modified Experiment (Part d)

Now, consider the spheres with different diameters. The air resistance coefficient is given as D = C_0 d^2. Rewriting the net force equation: ma = mg - C_0 d^2 v^2, and solving for acceleration: a = g - (C_0 d^2 v^2)/m.

06

- Mass in Terms of Diameter (Part d)

For solid spheres of the same material, the mass m is proportional to the volume, which depends on the cube of the diameter. Therefore, m = k(d^3), where k is a proportionality constant. Substituting this into the acceleration formula gives: a = g - (C_0 d^2 v^2)/(k d^3) = g - (C_0 v^2)/(k d).

07

- Compare Accelerations for Different Diameters (Part e)

From the formula a = g - (C_0 v^2)/(k d), observe that as the diameter d increases, the term (C_0 v^2)/(k d) decreases. Hence, the sphere with the larger diameter will have a smaller acceleration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Free-body Diagram

A free-body diagram is a visual representation of all the forces acting on an object. For a sphere falling through the air, two main forces are present:

The downward force of gravity, represented as \(mg\) (where \(m\) is the mass of the sphere and \(g\) is the acceleration due to gravity).

The upward force of air resistance, which is given by \(Dv^2\) according to the problem statement (where \(D\) is the drag coefficient and \(v\) is the velocity of the sphere).

By including these two forces in a free-body diagram, one can clearly see how the forces balance or don't balance, helping to understand the motion of the object.

Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the mass of that object times its acceleration (\( F = ma \)). When we apply this to a falling sphere with air resistance, we get:

\( ma = mg - Dv^2 \)

This equation illustrates that the net force is the difference between the gravitational force pulling the sphere downward and the air resistance pushing it upward. By simplifying this, we can derive the acceleration of the sphere as:

\( a = g - \frac{Dv^2}{m} \)

This formula shows that the acceleration diminishes as the air resistance increases.

Air Resistance

Air resistance, or drag, is a force that opposes the motion of an object through the air. In our problem, it's described by a quadratic law, \( Dv^2 \). This means that the air resistance grows with the square of the sphere's velocity.

As the sphere speeds up, the air resistance increases significantly, which eventually balances out the gravitational force. This balance leads to terminal velocity, the constant speed where acceleration stops because the forces are equal and opposite.

The coefficient \(D\) is directly related to the physical characteristics of the sphere, such as diameter when using \( D = C_0 d^2 \) as further explained in the modified experiment section.

Gravitational Force

Gravitational force is the force with which the Earth attracts objects toward its center. It is calculated by the formula:

\( F_g = mg \)

where \( m \) is the mass of the sphere and \( g \) is the acceleration due to gravity (approximately \( 9.8 \frac{m}{s^2} \).

This force is crucial in determining the initial acceleration of the sphere when it begins to fall. However, as velocity increases and air resistance starts to counteract this force, the net acceleration decreases until terminal velocity is reached. This delicate balance of forces explains the different accelerations and terminal velocities for objects of varying masses and sizes as shown in the problem.

Understanding gravitational force helps in visualizing why larger mass results in a greater net force and influence on the sphere's motion, reinforcing why heavier objects have distinct behaviors compared to lighter ones in the presence of air resistance.