Question

When a meter scale is balanced above a wedge, \(1 \mathrm{~kg}\) mass is hung at \(10 \mathrm{~cm}\) mark and a \(2 \mathrm{~kg}\) mass is hang at the \(85 \mathrm{~cm}\) mark. To which mark on the meter scale, the fulcrum be shifted (Neglect mass of meter scale) to balance the scale? (A) \(70 \mathrm{~cm}\) (B) \(50 \mathrm{~cm}\) (C) \(60 \mathrm{~cm}\) (D) \(65 \mathrm{~cm}\)

Short answer

The fulcrum should be shifted to the \(60 \mathrm{~cm}\) mark to balance the scale. The correct answer is (C) \(60 \mathrm{~cm}\).

Step-by-step solution

Identify the forces acting on the system

We have two weights hanging on the meter scale at different positions. The first weight is 1 kg at the 10 cm mark, and the second weight is 2 kg at the 85 cm mark.

Set up an equation for the torques

We will assume that the fulcrum is at position x cm from the left side of the meter scale. So, the distance between 1 kg mass and the fulcrum is (x - 10) cm, and the distance between 2 kg mass and the fulcrum is (85-x) cm. Since the system is balanced, the total torque on the meter scale due to both masses must be zero. Therefore, the torque due to the 1 kg mass must be equal and opposite in direction to the torque due to the 2 kg mass. \[ (1 \mathrm{~kg})(x - 10 \mathrm{~cm}) =( 2 \mathrm{~kg})(85-x \mathrm{~cm}) \]

Solve the equation for the position of the fulcrum

Now, we will solve the equation in step 2 for the position of the fulcrum (x). \[ x-10 = 170 - 2x \] By adding 2x to both sides and adding 10 to both sides, we get, \[ 3x = 180 \] Now, divide both sides by 3 to find the position of the fulcrum (x). \[ x = 60 \mathrm{~cm} \] So, the fulcrum should be shifted to the 60 cm mark to balance the scale. The correct answer is (C) \(60 \mathrm{~cm}\).

Key concepts

Lever Arm

When dealing with torque, the concept of the lever arm plays a crucial role. The lever arm, also known as the moment arm, is the perpendicular distance from the axis of rotation (in this case, the fulcrum) to the line of action of the force. In simpler terms, it's how far away from the pivot point a force is applied.
To visualize this, imagine a seesaw. The length of the board to the point where a child is sitting acts as the lever arm for that child's weight. The farther away from the center, the greater the force the child can exert to lift someone on the other end.
In our exercise, two masses hang on different points of a meter scale. The distance from where each mass hangs to the fulcrum is the lever arm. The first weight (1 kg at the 10 cm mark) has a lever arm of \(x - 10\) cm if the fulcrum is at position \(x\). Meanwhile, the second weight (2 kg at the 85 cm mark) has a lever arm of \(85 - x\) cm.
It's important to remember that the longer the lever arm, the more torque a force will produce. That's why changing the position of the fulcrum to 60 cm ensures both weights create equal and opposite torques, balancing the system.

Rotational Equilibrium

Rotational equilibrium occurs when the total net torque acting on a system is zero. This means there is no unbalanced rotational force, so the system remains static or rotates at a constant rate. For an object to be in rotational equilibrium, all the clockwise torques must balance out with all the counter-clockwise torques.
In the given problem, we have a meter scale balanced above a fulcrum. This setup will only maintain equilibrium if the torques exerted by each of the masses cancel each other out.

• Clockwise torque: Produced by the mass at the 10 cm mark. The formula is \(1 \times (x - 10)\) where \(x\) is the position of the fulcrum.
• Counter-clockwise torque: Produced by the mass at the 85 cm mark. This is calculated using \(2 \times (85 - x)\).

For the meter scale to be in rotational equilibrium, the equation \(1 \times (x - 10) = 2 \times (85 - x)\) must be satisfied. Solving this equation reveals that equilibrium is reached when the fulcrum is at the 60 cm mark, ensuring no net rotational movement.

Physics Problem Solving

Effective problem-solving in physics requires a step-by-step approach. Let's break this down with respect to the given exercise.

Understanding the Problem:

The first step in physics problem solving is to understand what is being asked. In this scenario, we need to determine the location of the fulcrum so that the meter scale is in balance. This requires establishing conditions for rotational equilibrium using the idea of torques.

Identifying Key Information:

Analyze the problem to understand the forces at play. Notice the masses and their positions along the scale. Recognize that the forces exerted by these masses affect the balance based on their distances from the fulcrum.

Setting Up Equations:

Use the relationships of lever arms and torques to set up mathematical equations. Here, we equate the torques produced by each mass to ensure they counterbalance each other.

Solving the Equations:

With the set equations, isolate the variable representing the unknown (fulcrum location) and solve it systematically.
This methodical approach helps simplify complex problems and emphasizes the power of algebraic manipulation in physics. Thus, by carefully analyzing each element involved and reliably using physical principles, students can find solutions efficiently.