Question

Arriving on a newly discovered planet, the captain of a spaceship performed the following experiment to calculate the gravitational acceleration for the planet: She placed masses of \(100.0 \mathrm{~g}\) and \(200.0 \mathrm{~g}\) on an Atwood device made of massless string and a frictionless pulley and measured that it took \(1.52 \mathrm{~s}\) for each mass to travel \(1.00 \mathrm{~m}\) from rest. a) What is the gravitational acceleration for the planet? b) What is the tension in the string?

Short answer

Answer: The gravitational acceleration on the planet is approximately 2.60 m/s², and the tension in the string is about 0.35 N.

Step-by-step solution

Calculate the effective acceleration (a)

Use the equation of motion \(d = \frac{1}{2}at^2\). Rearrange the equation to get \(a = \frac{2d}{t^2}\). Plug in d = 1.00 m and t = 1.52 s to get the effective acceleration. \(a = \frac{2 \times 1.00}{1.52^2} = \frac{2}{2.3104} \approx 0.8655\,\mathrm{m/s^2}\)

Calculate the gravitational acceleration (g)

Now, we can use the effective acceleration (a) to find the gravitational acceleration (g). First, we will calculate the effective force (F), which is the difference between the forces on the two masses (\(F = m_2g - m_1g\)). Using Newton's second law (\(F = ma\)), we can write this equation as \(m_2g - m_1g = (m_1 + m_2)a\). Rearrange the equation to get \(g = \frac{a(m_1+m_2) } {m_2 - m_1}\). Plug in m₁ = 100 g, m₂ = 200 g, and a = 0.8655 m/s², remembering to convert masses to kg. \(g = \frac{0.8655(0.1+0.2)} {0.2-0.1} \approx 2.5965\,\mathrm{m/s^2}\) The gravitational acceleration for the planet is approximately 2.60 m/s². #b)# We will now find the tension in the string (T). We can use either mass m₁ or m₂ to find the tension. In this case, we will use mass m₁.

Calculate the tension in the string using mass m₁

Use Newton's second law for mass m₁: \(T - m_1g = m_1a\). Rearrange the equation to get \(T = m_1g + m_1a\). Plug in m₁ = 100 g, g = 2.60 m/s², and a = 0.8655 m/s², remembering to convert mass to kg. \(T = 0.1 \times (2.60 + 0.8655) \approx0.34655\,\mathrm{N}\) The tension in the string is approximately 0.35 N.

Key concepts

Understanding the Atwood Machine

An Atwood machine is a simple yet fundamental device in classical mechanics, consisting of two masses connected by a string that goes over a pulley. When gravity pulls on these masses, they create two different forces due to their varying weights. Given that the string is massless and the pulley is frictionless, the net force leads to an acceleration of both masses.

This device beautifully illustrates how forces and motion interact and provides a hands-on approach to studying dynamics. It can be set up to provide a direct measure of gravitational acceleration by considering the motion of the masses and the physical characteristics of the system.

The device is often used in physics education to demonstrate fundamental principles such as Newton's second law of motion and to elucidate the role of tension in a string connecting two masses.

Newton's Second Law and Its Role

Newton's second law of motion is pivotal in calculating the dynamics of objects and is mathematically expressed as \( F = ma \), where \( F \) is the net force applied to the object, \( m \) is the mass, and \( a \) is the acceleration.

In the context of the Atwood machine, this law lets us equate the net force acting on the system with the effective acceleration of the masses. It connects the tension in the string and the gravitational pull on each mass, allowing the calculation of the system's behavior, including individual forces and overall acceleration.

By rearranging Newton's second law, we can isolate different physical quantities, such as tension or gravitational acceleration, which are crucial to understanding the system's mechanics.

Deciphering Effective Acceleration

The term 'effective acceleration' refers to the observed acceleration of masses in a system like the Atwood machine. It's not equal to gravitational acceleration because the system's dynamics involve two masses with a shared acceleration and the presence of tension in the string.

To calculate the effective acceleration, we use kinematic equations that relate displacement, time, and acceleration. In our spaceship captain's experiment, the formula \( a = \frac{2d}{t^2} \) provides the value of the effective acceleration based on the measured time and distance. This acceleration is key to uncovering the value of the gravitational acceleration on the new planet.

Effective acceleration incorporates all the forces at play, giving us a net acceleration figure that reflects the actual motion of the system.

The Significance of Tension in the String

Tension in the string plays a subtle but crucial role in the dynamics of an Atwood machine. Since the string is considered massless and the pulley frictionless, the tension is the same throughout the entire string. This tension ensures that both masses are interdependent in their motion.

By applying Newton's second law to one of the masses, we can determine the tension. The equation \( T = m_1g + m_1a \) uses the mass, the gravitational acceleration, and the effective acceleration. The calculation of tension is not just a theoretical exercise; it gives insights into how real-world factors like string strength and mass limits can affect the performance of systems where tension is a critical factor.