Question

A certain particle has a weight of \(26.0 \mathrm{~N}\) at a point where the acceleration due to gravity is \(9.80 \mathrm{~m} / \mathrm{s}^{2} .(a)\) What are the weight and mass of the particle at a point where the acceleration due to gravity is \(4.60 \mathrm{~m} / \mathrm{s}^{2} ?(b)\) What are the weight and mass of the particle if it is moved to a point in space where the gravitational force is zero?

Short answer

The weight and mass of the particle at a point where the acceleration due to gravity is \(4.60 \, m/s^2\) are 12.2 N and 2.65 kg. When the particle is moved to a point in space where the gravitational force is zero, its weight will be 0 N, but its mass will remain 2.65 kg.

Step-by-step solution

Calculate the initial mass

Start by calculating the initial mass of the particle using the given weight and gravity at the original point. The formula for weight is \( W = m \cdot g \), where \( W \) is weight, \( m \) is mass and \( g \) is the acceleration due to gravity. This allows to solve for mass as follows: \( m = W / g \). With the given values (weight \( W = 26.0 \, N \) and gravity \( g = 9.80 \, m/s^2 \)), the mass becomes \( m = 26.0 \, N / 9.80 \, m/s^2 = 2.65 \, kg \).

Calculate the new weight

To find the new weight of the particle at a point where gravity is \( g = 4.60 \, m/s^2 \), use the weight formula \( W = m \cdot g \) again. With the mass \( m = 2.65 \, kg \) and gravity \( g = 4.60 \, m/s^2 \), the new weight becomes \( W = 2.65 \, kg \cdot 4.60 \, m/s^2 = 12.2 \, N \).

Calculate weight and mass at zero gravity

If a particle is moved to a point where the gravitational force is zero, the weight of the object will be zero as well because weight is the force exerted on an object due to gravity. The mass of the particle remains the same and is not affected by the location in the universe, so it stays \( m = 2.65 \, kg \).

Key concepts

Acceleration Due to Gravity

When we discuss the concept of acceleration due to gravity, we're referring to the rate at which an object accelerates towards the center of a massive body, like the Earth, due to gravitational force. This acceleration is represented by the symbol 'g' and on Earth's surface, it has an approximate value of \(9.81 \text{ m/s}^2\). However, 'g' can vary depending on where you are in the universe — it's less on the surface of the Moon, and it can be zero in the vast reaches of space where the gravitational influence of celestial bodies is negligible.

Understanding how 'g' influences weight is crucial. 'Weight' is the force exerted on an object due to gravity and is calculated by the formula \( W = m \times g \), where 'W' stands for weight, 'm' for mass, and 'g' for acceleration due to gravity. For example, if you were to move from Earth to a planet with less gravity, even though your mass remains the same, you would weigh less because 'g' is smaller.

Mass and Weight Relationship

The terms 'mass' and 'weight' are often used interchangeably, but they represent very different concepts. Mass is a measure of the amount of matter in an object, usually measured in kilograms or grams. It's an intrinsic property, meaning it doesn't change regardless of location. Weight, on the other hand, is the force that gravity exerts on an object's mass and it can change depending on the gravitational pull at a location.

To determine an object's weight from its mass, we use the equation mentioned earlier: \( W = m \times g \). This means that weight is directly proportional to both the mass of the object and the acceleration due to gravity at that location. If you take a 2.65 kg object to a place with no gravitational pull, its mass remains 2.65 kg, but its weight drops to zero Newtons (N), because weight depends on the presence of a gravitational field.

Newton's Laws of Motion

Sir Isaac Newton's laws of motion form the foundation for classical mechanics and explain the relationship between the motion of an object and the forces acting upon it.

• The first law, also known as the law of inertia, states that an object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
• The second law establishes the equation \( F = m \times a \), stating that the force applied to an object is equal to the mass of the object times its acceleration. This law provides a calculation for weight as a type of force: an object's mass multiplied by the acceleration due to gravity equals its weight.
• The third law is known as the action-reaction law, which states that for every action, there is an equal and opposite reaction.

These laws are pivotal in understanding physics-based problems, such as calculating how different gravitational strengths affect the weight of an object. By applying these fundamental principles, we can solve a wide range of problems in physics and beyond.