Problem 4 - College Physics

Question

A net force of 4.0 N gives an object an acceleration of \(10 \mathrm{~m} / \mathrm{s}^{2} .\) What is the mass of the object?

Step-by-Step Solution

Answer

The mass of the object is 0.4 kg.

Step 1: Understanding the Formula

To find the mass of an object when the net force and acceleration are given, use Newton's second law, which states that force equals mass times acceleration, or \( F = ma \). Here, \( F \) is force, \( m \) is mass, and \( a \) is acceleration.

Step 2: Isolating the Mass in the Formula

Rearrange the formula \( F = ma \) to solve for mass \( m \). This can be done by dividing both sides of the equation by acceleration \( a \). The formula now becomes \[ m = \frac{F}{a} \].

Step 3: Substituting the Known Values

Substitute the given values into the equation: \( F = 4.0 \) N and \( a = 10 \) m/s\(^2\). Thus, the equation becomes \[ m = \frac{4.0 \text{ N}}{10 \text{ m/s}^2} \].

Step 4: Performing the Calculation

Calculate the value by dividing 4.0 by 10. \[ m = 0.4 \text{ kg} \]. This result gives the mass of the object.

Key Concepts

Understanding Force

Force is a fundamental concept in physics and is defined as any interaction that, when unopposed, changes the motion of an object. It is a vector quantity, which means it has both a magnitude and a direction. The unit of force in the International System of Units (SI) is the Newton (N).

• 1 Newton is the force required to accelerate a 1-kilogram mass by 1 meter per second squared (1 N = 1 kg·m/s²).
• Forces can be caused by phenomena such as gravity, friction, or applied push or pull.

When multiple forces act on an object, they combine to form a net force, which determines the object’s overall acceleration. If the forces are balanced, the net force is zero, and the object remains in its current state of motion. In our example, a net force of 4 N is acting on the object.

Exploring Acceleration

Acceleration refers to the rate of change of velocity of an object. It is also a vector quantity, having both magnitude and direction. When an object speeds up, slows down, or changes direction, it is said to be accelerating.

• Acceleration is measured in meters per second squared (m/s²).
• Positive acceleration indicates an increase in velocity, while negative acceleration (deceleration) signifies a decrease in velocity.

In the given exercise, the object has an acceleration of 10 m/s². This means that for each second, the object’s velocity changes by 10 meters per second, illustrating how the force directly impacts the object's motion.

The Role of Mass

Mass is a measure of the amount of matter in an object and does not change regardless of its location in the universe. Unlike weight, which is affected by gravity, mass remains constant. Mass is a scalar quantity, meaning it has magnitude but no direction.

• It is measured in kilograms (kg) in the SI system.
• Mass influences how much force is needed to produce a certain acceleration.

Newton's second law describes the relationship between mass and acceleration, indicating that greater mass requires more force to achieve the same acceleration. In our exercise, we calculate the mass of an object given the force and acceleration using Newton's formula.

Executing the Calculation

To find the mass using the given force and acceleration, we apply Newton's second law of motion, which is expressed as:\[ F = ma \]Since we need to solve for mass, rearrange the formula:\[ m = \frac{F}{a} \]Using the provided values:

• Net force \( F \) is 4.0 N
• Acceleration \( a \) is 10 m/s²

Substitute these into the equation:\[ m = \frac{4.0 \text{ N}}{10 \text{ m/s}^2} \]By performing the calculation, dividing 4.0 by 10, we find:

• \( m = 0.4 \text{ kg} \)

Thus, the mass of the object is 0.4 kilograms, allowing us to understand how the force and acceleration data leads to this result.