Question

4.39 Arriving on a newly discovered planet, the captain of a spaceship performed the following experiment to calculate the gravitational acceleration for the planet: He 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 s for each mass to travel \(1.00 \mathrm{~m}\) from rest.

Short answer

Answer: The gravitational acceleration for the newly discovered planet is approximately 9.36 m/s^2.

Step-by-step solution

Understanding the given data

In the given exercise, we have an Atwood device with two masses m1 = 100 g and m2 = 200 g. It takes 1.52 s for each mass to travel 1.00 m from rest. We are asked to find the gravitational acceleration of the planet (g).

Converting given masses to kg

First, let's convert the given masses to kg: m1 = 100 g = 100/1000 kg = 0.1 kg m2 = 200 g = 200/1000 kg = 0.2 kg

Applying Newton's second law of motion

Let's apply Newton's second law of motion to the masses of the Atwood device: For mass m1: T - m1 * g = m1 * a (1) For mass m2: m2 * g - T = m2 * a (2) Add equation (1) and (2) to eliminate the tension T, we get: m2 * g - m1 * g = (m1 + m2) * a Find a in terms of g: a = (m2 - m1) * g / (m1 + m2)

Using kinematic equation

We know the distance traveled (s = 1 m) and the time taken (t = 1.52 s). We can use the following kinematic equation to relate distance, acceleration, and time: s = ut + 0.5 * a * t^2 where u is the initial velocity and is given as 0 since the masses start from rest. Plug in the values, 1 = 0.5 * a * (1.52)^2

Solving for g

Now, we have: 1=0.5*(m2-m1) *g/(m1+m2)*(1.52)^2 Rearrange the equation to solve for g, g = 2 * 1 / [(1.52)^2 * (m2 - m1) / (m1 + m2)] Substitute the values for m1 and m2, g = 2 * 1 / [(1.52)^2 * (0.2 - 0.1) / (0.1 + 0.2)] Using a calculator, we get, g ≈ 9.36 m/s^2 The gravitational acceleration for the newly discovered planet is approximately 9.36 m/s^2.

Key concepts

Atwood Device

An Atwood device is a simple yet fundamental apparatus used to study motion and forces in physics. It is essentially composed of two masses attached to either end of a string that passes over a pulley. The beauty of this device lies in its ability to illustrate and measure the effects of gravity on two objects of different masses.

When the apparatus is set into motion, the heavier mass will accelerate downwards while the lighter one moves upwards, at the same rate. This movement enables us to explore the relationships between mass, acceleration, and gravitational forces. While the pulley is assumed to be frictionless and the string massless in theoretical scenarios, these considerations are critical for achieving accurate results in practical experiments.

For those looking to understand the mechanics of motion, the Atwood device serves as a clear representation of Newton's laws at work. It also provides a platform to exercise problem-solving skills by applying kinematic equations and principles from dynamics to calculate unknown variables such as gravitational acceleration, as seen in the textbook exercise we are examining.

Newton's Second Law of Motion

Newton's second law of motion is a cornerstone of classical mechanics that establishes a quantitative relationship between force, mass, and acceleration. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration \( F = m \cdot a \). This principle is eloquently simple but incredibly powerful in describing how the forces acting on an object change its motion.

In the context of the Atwood device, Newton's second law allows us to analyze the forces acting on each mass. Because the system of the two masses and the pulley is subject to external gravitational force, we can apply this law to calculate the net force and subsequently the acceleration of the system. Once we know the acceleration, determining the gravitational acceleration \(g\) on the new planet becomes feasible.

Understanding this law is vital when we deal with varying forces and different masses, as it provides a direct method to quantify movement and force. It is particularly useful in this exercise challenge where understanding motion under the influence of gravity is required.

Kinematic Equations

Kinematic equations are a set of formulas in physics that describe the motion of objects without considering the forces that cause the motion. These equations relate five kinematic variables: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). Often in an introductory physics course, you will work with the simplified scenario where the acceleration is constant.

In the context of the Atwood device, we use a kinematic equation to link the distance one of the masses travels to the time it takes and the acceleration it undergoes. The specific equation used in our exercise is \( s = ut + \frac{1}{2} a t^2 \), where \(s\) represents the distance traveled, \(u\) is the initial velocity (which is zero in this case because the masses start from rest), \(a\) is the acceleration, and \(t\) is the time.

By manipulating these equations properly, we can solve for one variable when the others are known. Kinematic equations are not just theoretical; they align perfectly with real-world applications like calculating how fast a car needs to go to come to a stop within a certain distance or, as in our exercise, finding the gravitational acceleration on a new planet.

Converting Mass Units

In physics problems, especially in the field of mechanics, mass is a fundamental quantity. However, mass can be measured in various units, and it is crucial to use the correct unit for calculations. Converting mass units is a simple, yet essential skill when working with equations that involve mass.

In our scenario with the Atwood device, the given masses are in grams (g), but the standard unit for mass in the International System of Units (SI) is kilograms (kg). To convert grams to kilograms, we divide by 1,000 because there are 1,000 grams in one kilogram \(1 \text{kg} = 1,000 \text{g}\).

For example, to convert the 100 g and 200 g to kilograms, we simply perform the following:

• For \(m1\): \(100 \text{g} = \frac{100}{1,000} \text{kg} = 0.1 \text{kg}\)
• For \(m2\): \(200 \text{g} = \frac{200}{1,000} \text{kg} = 0.2 \text{kg}\)

Converting mass units correctly is imperative to get accurate results, as seen in the steps to solve the gravitational acceleration for the newly discovered planet. Without such conversions, the values derived from the equations would not be consistent with SI units, potentially leading to incorrect conclusions.