Problem 36 - Beginning and Intermediate Algebra

Question

The force exerted on an object varies jointly as the mass and acceleration of the object. If a 20 -Newton force is exerted on an object of mass \(10 \mathrm{kg}\) and an acceleration of \(2 \mathrm{m} / \mathrm{sec}^{2},\) how much force is exerted on a \(50 \mathrm{kg}\) object with an acceleration of \(8 \mathrm{m} / \mathrm{sec}^{2} ?\)

Step-by-Step Solution

Answer

The force exerted on the 50 kg object with an acceleration of 8 m/s^2 is 400 N.

Step 1: Write the given information in equations for both objects

In this step, we will write the given information about the first object (20 N force, 10 kg mass, and 2 m/s^2 acceleration) and the second object (unknown force F, 50 kg mass, and 8 m/s^2 acceleration) using Newton's second law \(F = ma\): For the first object: \(20 \mathrm{N} = 10 \mathrm{kg} * 2 \frac{\mathrm{m}}{\mathrm{s^2}}\) For the second object: \(F = 50 \mathrm{kg} * 8 \frac{\mathrm{m}}{\mathrm{s^2}}\)

Step 2: Solve for F in the second object's equation

In this step, we will apply Newton's second law to calculate the force exerted on the second object by multiplying its mass and acceleration: \(F = 50 \mathrm{kg} * 8 \frac{\mathrm{m}}{\mathrm{s^2}}\) \(F = 400 \mathrm{N}\)

Step 3: State the final answer

The force exerted on the 50 kg object with an acceleration of 8 m/s^2 is 400 N.

Key Concepts

Force Calculation

Calculating force involves understanding how different elements like mass and acceleration work together. According to Newton's Second Law of Motion, force is calculated using the equation:\[ F = ma \]where:

• \( F \) is the force exerted on an object, measured in Newtons (N).
• \( m \) is the mass of the object, measured in kilograms (kg).
• \( a \) is the acceleration of the object, measured in meters per second squared (m/s²).

This formula shows that force is the product of mass and acceleration. This means that both mass and acceleration directly impact the force: increase either and the force increases too. For example, if you have a mass of 10 kg and accelerate it at 2 m/s², the force would be \( 20 \) N.

Joint Variation

Joint variation refers to a situation where a variable depends on two or more other variables directly. When we say that force varies jointly with mass and acceleration, it means that both factors together determine how much force will be exerted. When either the mass or the acceleration increases, the force increases as well—as long as the other remains constant. For instance, in our exercise, the force on an object was updated from a 20-N on a smaller object to 400 N on a larger, faster-moving object, illustrating joint variation. Understanding this concept helps in predicting how changes in mass and acceleration affect force.

Mass and Acceleration

Mass and acceleration are key components in determining the force applied to an object. Mass is a measure of how much matter is in an object, and it affects how much force is needed to move it. Acceleration describes how quickly an object is changing its speed or direction. In Newton's formula, these two properties work together:

• The larger the mass, the more force is required to achieve the same acceleration.
• The greater the acceleration applied to a mass, the greater the force exerted.

For instance, multiplying the mass of 50 kg by an acceleration of 8 m/s² gives a force of 400 N. This exemplifies how increasing both mass and acceleration can significantly boost the exerted force.