Question

In a physics class, a 2.70 - g ping pong ball was suspended from a massless string. The string makes an angle of \(\theta=15.0^{\circ}\) with the vertical when air is blown horizontally at the ball at a speed of \(20.5 \mathrm{~m} / \mathrm{s}\). Assume that the friction force is proportional to the squared speed of the air stream. a) What is the proportionality constant in this experiment? b) What is the tension in the string?

Short answer

Solution: The proportionality constant (k) is 6.1056 x 10^-4 Ns²/m², and the tension in the string (T) is 0.02564 N.

Step-by-step solution

Identify the forces acting on the ping pong ball

There are three forces acting on the ping pong ball: - Weight (W) = m * g, where m is the mass of the ping pong ball, and g is the acceleration due to gravity. - Tension (T) = force acting along the string, in the same direction as the string. - Friction force (F) = k * v², where k is the proportionality constant, and v is the speed of the air stream.

Write equations of motion

The weight can be divided into two components: one parallel to the string tension (along the y-axis) and another perpendicular to the tension (along the x-axis). We'll label these components W_y and W_x, respectively. W_y = W * cos(θ) W_x = W * sin(θ) Now, we'll write the equations of motion using Newton's second law: For horizontal direction (x-axis): F - W_x = m * a_x k * v² - W * sin(θ) = m * a_x (1) For vertical direction (y-axis): T - W_y = m * a_y T - W * cos(θ) = m * a_y (2) Since the ping pong ball remains in equilibrium, there is no net acceleration in the x and y directions (both a_x and a_y are 0).

Solve the equations simultaneously

For equation (1): k * v² - W * sin(θ) = m * a_x = 0 k = (m * g * sin(θ)) / v² Substitute the given values: k = ((2.7 x 10^-3 kg) * (9.81 m/s²) * sin(15°)) / (20.5 m/s)² k = 6.1056 x 10^-4 Ns²/m² For equation (2): T - W * cos(θ) = m * a_y = 0 T = W * cos(θ) T = m * g * cos(θ) Substitute the given values: T = (2.7 x 10^-3 kg) * (9.81 m/s²) * cos(15°) T = 0.02564 N So, the proportionality constant (k) is 6.1056 x 10^-4 Ns²/m², and the tension in the string (T) is 0.02564 N.

Key concepts

Proportionality Constant

In physics, the proportionality constant plays a crucial role in understanding the relationships between variables where one variable is proportional to another. In the context of frictional forces like the one described in the exercise, the proportionality constant (denoted as 'k') connects the force of friction to the square of the speed of the air stream blowing past the ping pong ball.

The formula for the friction force is given by
\( F = k \times v^2 \),
where 'F' is the friction force, 'v' is the speed of the air, and 'k' is the proportionality constant we want to determine. Intuitively, 'k' signifies how strongly the speed of the air affects the magnitude of the force acting on the ping pong ball. A larger 'k' value would mean a greater force at any given speed, suggesting the surface or situation offers more resistance to motion.

Tension in a String

Tension is the force conducted along the length of a string or rope when it is subjected to forces at its ends. It's important to recognize that tension has both magnitude and direction, and these properties are transmitted through the string. In our ping pong ball exercise, the tension in the string not only supports the weight of the ball but also counteracts the horizontal friction force.

Tension is calculated by considering the forces along the string's direction. Since the ball is in equilibrium and not accelerating, the net force in any direction must be zero. The equation
\( T = m \times g \times \text{cos}(\theta) \)
shows that the tension equals the vertical component of the ball's weight. Through this concept, learners can better understand how forces are balanced in static situations, and how tension is adjusted based on the angle of the string.

Equations of Motion

Equations of motion are core concepts in classical mechanics that describe the relationships between the motion of an object and the forces applied to it. Newton's second law, stated as
\( F = m \times a \)
is a cornerstone of these equations, linking the net force acting on an object (F) to its mass (m) and the acceleration (a) it experiences.

In scenarios like the hanging ping pong ball, equations of motion are critical for describing how the ball behaves when subjected to multiple forces. By recognizing the ball's equilibrium where acceleration is zero, we see that the forces along each axis (horizontal and vertical) must sum to zero. This insight is essential for solving problems involving the motion of objects, whether they are accelerating or at rest, providing a clear method for quantifying the impact of forces.

Force Analysis

Force analysis is part of a broader physics study concerning the effects of forces on the movement and equilibrium of objects. It includes identifying all the forces acting on an object and using Newton's laws to describe the motion or stationary state of the object. Understanding force analysis allows students to predict how objects will react under various conditions.

When dealing with the suspended ping pong ball, we analyze the weight acting downward and its components along the x and y axes, the tension in the string, and the horizontal friction force. The ball's equilibrium leads us to conclude that the sum of forces in both x and y directions must be zero, laying the foundation for calculating unknown variables like the tension and the proportionality constant. This step-by-step approach underpins much of classical mechanics and engineering problems, among other fields.