Question

A 15-kg child sits on a playground seesaw, 2.0 m from the pivot. A second child located 1.0 m on the other side of the pivot would have to have a mass ________ to lift the first child off the ground.

Short answer

Solution: Step 1: Calculate the torque exerted by the first child T1 = (15 kg × 9.81 m/s²) × 2.0 m T1 = 294.3 kg·m²/s² Step 2: Rewrite T2 in terms of m2 T2 = (m2 × 9.81 m/s²) × 1.0 m Step 3: Set the torques equal and solve for m2 (294.3 kg·m²/s²) = (m2 × 9.81 m/s²) × 1.0 m Step 4: Find the mass of the second child required to lift the first child m2 = (15 kg × 2.0) m2 = 30 kg Answer: The mass of the second child required to lift the first child off the ground is 30 kg.

Step-by-step solution

Understand and calculate torque for the first child

Since the seesaw is balanced when the torques exerted by both children are equal, we need to find out the torque exerted by the first child. The force exerted by the first child is the gravitational force acting downwards: Force = mass × acceleration due to gravity (F1 = m1 × g) where m1 = 15 kg (mass of first child) and g = 9.81 m/s² (acceleration due to gravity). The torque is the product of force and distance from the pivot: Torque = Force × distance (T1 = F1 × d1) Plugging the values, we have: T1 = (15 kg × 9.81 m/s²) × 2.0 m Calculate T1.

Calculate the force exerted by the second child

We know that torque exerted by both children should be equal for the seesaw to be balanced: T1 = T2. Therefore, we can write the equation for the torque of the second child as: T2 = F2 × d2 where F2 is the force exerted by the second child and d2 is the distance from the pivot (1.0 m). But first, let's find F2 in terms of the mass of the second child (m2). F2 = m2 × g Now rewrite T2 in terms of m2: T2 = (m2 × 9.81 m/s²) × 1.0 m

Set the torques equal and solve for m2

Since T1 = T2, we can set the torque equations equal and solve for m2: (15 kg × 9.81 m/s²) × 2.0 m = (m2 × 9.81 m/s²) × 1.0 m Divide both sides by 9.81 m/s²: (15 kg × 2.0 m) = (m2 × 1.0 m) Now, divide both sides by 1.0 m: (15 kg × 2.0) = m2 Finally, calculate m2.

Find the mass of the second child required to lift the first child

Having found m2, the mass of the second child required to lift the first child off the ground is m2 = (15 kg × 2.0) Calculate m2, which is the mass of the second child.

Key concepts

Torque and Equilibrium

Understanding torque and equilibrium are fundamental to analyzing situations like seesaw balance. Torque, also known as the moment of force, is a measure of the rotational force that causes an object to turn or twist. On a seesaw, this rotational force is created by the weight of the children, acting a certain distance from the pivot, known as the moment arm.

In terms of physics, for an object to be in equilibrium, the total torque exerted on it must be zero. This condition means that a seesaw is balanced when the torque produced by one child on one side is equal to the torque produced by the second child on the other side. Mathematically, we can express this condition as \(T_1 = T_2\), where \(T_1\) is the torque from the first child and \(T_2\) is the torque from the second child.

To achieve equilibrium on a seesaw, one can adjust either the distances (d) from the pivot or the masses (m) of the children—since torque is the product of the distance from the pivot and the force due to gravity on the mass sitting at that distance. Clear understanding of these principles allows us to solve seesaw physics problems accurately.

Gravitational Force

The gravitational force is the attraction between two masses, such as a child on a seesaw and the Earth. This force, which we commonly refer to as weight, is fundamental in seesaw problems. The equation that defines gravitational force is \(F = m \times g\), where \(m\) represents mass and \(g\) is the acceleration due to gravity, which on Earth's surface is approximately 9.81 m/s².

Each child on a seesaw experiences this force pulling them towards the ground, which is what generates the downward push on the seesaw. Since gravitational force is proportional to mass, a heavier child exerts a greater force, which in turn creates a larger torque when multiplied by the distance from the pivot. Recognizing this force and how it interacts with mass and distance is instrumental in solving for unknown variables in seesaw scenarios.

Moment of Force

Moment of force, interchangeably known as torque, is a concept that encompasses the rotational effect of a force applied at a distance from an axis or pivot. In the context of a seesaw, the moment arm is the distance from where the child is sitting to the pivot point. The moment of force is calculated by the equation \(Torque = Force \times distance\), or \(T = F \times d\).

It is critical to discern that the effectiveness of the force in causing rotation depends not only on the amount of force but also on where the force is applied. For example, a lighter child can balance a heavier child by sitting further from the pivot point. This principle underlies the strategy for determining the mass of the second child required to balance the seesaw in the given exercise. We must ensure that the product of force (caused by gravity) and distance (from the pivot) are equal for both children, effectively utilizing the concept of moment of force to solve for equilibrium.