Problem 1 - Applied Mathematics

Question

A particle of mass \(5 \mathrm{~kg}\) is pulled along a smooth horizontal surface by a horizontal string. The acceleration of the particle is \(10 \mathrm{~ms}^{-2}\). The tension in the string is: (a) \(2 \mathrm{~N}\) (b) \(50 \mathrm{~N}\) (c) \(5 \mathrm{~N}\) (d) \(15 \mathrm{~N}\) (e) \(10 \mathrm{~N}\).

Step-by-Step Solution

Answer

The tension in the string is 50 N (option b).

Step 1: Understand the Given Information

A particle with mass (m) of 5 kg is pulled on a smooth horizontal surface, meaning there is no friction. The acceleration (a) of the particle is given as 10 m/s². The goal is to find the tension (T) in the string.

Step 2: Use Newton’s Second Law

Newton’s second law states that the force (F) acting on an object is the product of its mass and acceleration. The formula is: \[F = m \times a\]. Since the tension in the string (T) is the force causing the particle to accelerate, we substitute T for F.

Step 3: Substitute the Known Values

Substitute the mass (m = 5 kg) and the acceleration (a = 10 m/s²) into the formula: \[T = 5 \times 10\].

Step 4: Calculate the Tension

Now calculate the value: \[T = 50\]. Thus, the tension in the string is 50 N.

Step 5: Select the Correct Answer

Among the given options, the correct option is (b) 50 N.

Key Concepts

Force Calculation

When dealing with physics problems that involve forces, understanding how to calculate force is crucial. Newton's Second Law provides a simple yet powerful equation to do this: \[ F = m \times a \]In this formula,

• F represents the force applied to an object,
• m stands for mass, and
• a is the acceleration.

For the given problem, the force we need to calculate is the tension in the string (T). The problem tells us the mass (m) of the particle is 5 kg and the acceleration (a) is 10 m/s². By substituting these values into the formula, we get:\[ T = 5 \text{ kg} \times 10 \text{ m/s}^2 = 50 \text{ N} \]So, the tension in the string is 50 N. This demonstrates how Newton's Second Law can be used to find forces based on known values of mass and acceleration. It is also essential for solving various mechanics problems in physics.

Mass and Acceleration

Understanding the relationship between mass and acceleration is key to comprehending Newton's Second Law. In the formula \[ F = m \times a \]mass (m) and acceleration (a) are directly proportional to the force (F). This means that:

• If you increase the mass while keeping acceleration constant, the force will increase proportionally.
• Similarly, if you increase the acceleration while keeping the mass constant, the force will also increase proportionally.

Let's take the exercise we worked on: a particle of 5 kg mass accelerates at 10 m/s². Here, you can see how changes in either mass or acceleration affect the result.If the mass was heavier, for instance, 10 kg, the force would double:\[ F = 10 \text{ kg} \times 10 \text{ m/s}^2 = 100 \text{ N} \]And if the acceleration was different, say 20 m/s², the force would also increase:\[ F = 5 \text{ kg} \times 20 \text{ m/s}^2 = 100 \text{ N} \]Recognizing this relationship helps you predict how forces will change when either mass or acceleration changes, which is a fundamental concept in dynamics.

Tension in Physics Problems

Tension is a common force that appears in many physics problems, especially those involving ropes, strings, or cables that pull or hang objects. In our exercise, tension is the force that pulls the particle and causes it to accelerate. To find the tension in the string, we used Newton's Second Law.Tension force often involves the following steps:

• Identify the forces acting on the object: In our problem, the only force considered was the tension in the string pulling the particle horizontally.
• Apply Newton’s Second Law: We write the equation of motion using the known values of mass and acceleration.
• Solve for tension: By substituting the known quantities into the equation, we calculated that tension, T, is 50 N.

Tension problems may vary, including vertical pulls or more complex setups with multiple objects. However, the process remains similar: identify forces, apply Newton's laws, and solve for the unknown. This step-by-step approach makes understanding tension in physics problems manageable and less daunting.