Question

A \(30 \mathrm{kg}\) girl sits on a seesaw at a distance of \(2 \mathrm{m}\) from the fulcrum. Where must her father sit to balance the seesaw if he has a mass of \(90 \mathrm{kg} ?\) (A) 67 cm from the girl (B) 67 cm from the fulcrum (C) 133 cm from the girl (D) 267 cm from the fulcrum

Short answer

The father must sit 67 cm from the fulcrum.

Step-by-step solution

- Understand the problem

We need to find the position where the girl's father should sit to balance the seesaw. The seesaw will be balanced when the clockwise moment (torque) about the fulcrum is equal to the counterclockwise moment (torque) about the fulcrum.

- Write the torque equation

The torque is given by the formula \[ \tau = F \times d \] where \[ F \] is the force (weight) and \[ d \] is the distance from the fulcrum. Since weight is the product of mass and gravity, we have: \[ \tau = m \times g \times d \]For the girl, \[ \tau_{girl} = 30 \times g \times 2 \]For the father, \[ \tau_{father} = 90 \times g \times d_{father} \]

- Set up the balance equation

To balance the seesaw, the torques must be equal:\[ 30 \times g \times 2 = 90 \times g \times d_{father} \]

- Solve for the father's distance

We can cancel \[ g \] from both sides of the equation:\[ 30 \times 2 = 90 \times d_{father} \ 60 = 90 \times d_{father} \ d_{father} = \frac{60}{90} \ d_{father} = \frac{2}{3} \mathrm{m} \]Convert \[ \frac{2}{3} \text{ meters} \] to centimeters: \[ \frac{2}{3} \times 100 = 66.67 \text{ cm} \approx 67 \text{ cm} \]

- Determine the correct option

The father must sit 67 cm from the fulcrum. Comparing options, the correct answer is (B) 67 cm from the fulcrum.

Key concepts

Torque

Torque, often referred to as the 'moment of force,' is a measure of the rotational force applied to an object about a pivot or fulcrum. Imagine trying to turn a nut with a wrench: the further your hand is from the nut (the fulcrum), the easier it is to turn. This is because torque depends on both the force you apply and the distance from the fulcrum. The formula for torque is \[ \tau = F \times d \] where: \[ \tau \] = torque, \[ F \] = force (weight in our case), and \[ d \] = distance from the fulcrum. The girl's weight creates a counterclockwise torque, while her father's weight would create a clockwise torque. To balance the seesaw, these torques must be equal.

Force Equilibrium

For any object to be in rotational equilibrium, the sum of all torques acting on it must be zero. On a balanced seesaw, the counterclockwise torque caused by one person must be equal to the clockwise torque caused by another. This state is known as 'force equilibrium.' In our exercise, the seesaw is balanced when the girl's torque is equal to her father's torque: \[ 30 \times g \times 2 = 90 \times g \times d_{father} \] This equation states that the weight (mass times gravity) and distance from the fulcrum must balance out for both individuals. Cancelling g out of the equation simplifies our calculations.

Moment of Force

The moment of force is another term for torque, emphasizing the turning effect that a force has when applied at a distance from a pivot point. It combines the linear force with the lever arm distance. This can be visualized with the seesaw: the further from the fulcrum a force (like weight) is applied, the greater the turning effect or moment. In our seesaw problem, the girl provides a certain moment of force due to her weight and distance from the fulcrum. To balance this, her father needs to create an equivalent moment of force on the opposite side by positioning himself appropriately. This balance ensures that the seesaw remains level.

Mass and Weight Calculation

Mass and weight are related but distinct concepts. Mass is the amount of matter in an object, typically measured in kilograms. Weight, on the other hand, is the force exerted by gravity on that mass. It is calculated as \[ weight = mass \times gravity \]. In our seesaw problem, we used the masses of the girl and her father (30 kg and 90 kg, respectively) to determine the forces they exert: the weight. The seesaw balances by ensuring the torques (moment of forces) from these weights are equal. This involves calculating the weight based on mass and then finding the right distance to balance out these forces.