Question

A cord, with a spring balance to measure forces attached midway along, is hanging from a hook attached to the ceiling. A mass of \(10 \mathrm{kg}\) is hanging from the lower end of the cord. The spring balance indicates a reading of \(98 \mathrm{N}\) for the force. Then two people hold the opposite ends of the same cord and pull against each other horizontally until the balance in the middle again reads \(98 \mathrm{N} .\) With what force must each person pull to attain this result?

Short answer

Answer: Each person must exert a force of 49 N to make the spring balance read 98 N.

Step-by-step solution

Determine the initial force on the spring balance

In the beginning, the mass of 10 kg is hanging vertically from the lower end of the cord. The force exerted by the mass on the spring balance is given by the product of the mass and the acceleration due to gravity (approximately 9.8 m/s^2). Therefore, Force exerted by the mass = \(10\,\textup{kg} \times 9.8\, \textup{m/s}^2 = 98\, \textup{N}\) Since the spring balance reads 98 N, the force acting on it is indeed 98 N.

Determine the forces on the spring balance while being pulled horizontally

In the second scenario, two people hold the opposite ends of the cord and pull it horizontally until the spring balance reads 98 N. We are supposed to find the force exerted by each person. Let's assume each person pulls with a force of F.

Use the principle of superposition

According to the principle of superposition, the resultant force acting on the spring balance in the second scenario should also be equal to 98 N. When two forces of equal magnitude F act at 180 degrees from each other, the resultant force is given by: Resultant force = \(F + F = 2F\) Since the spring balance reads 98 N, we have: \(2F = 98\, \textup{N}\)

Calculate the force exerted by each person

Now, we will solve the equation to find the force exerted by each person: \(F = \frac{98\, \textup{N}}{2} = 49\, \textup{N}\) So, each person must pull the cord with a force of 49 N to attain a reading of 98 N on the spring balance.