Problem 1 - AP Physics 1 with 2 Practice Test 2nd

Question

A \(45-\mathrm{kg}\) girl is sitting on a seesaw \(0.6 \mathrm{~m}\) from the balance point, as shown below. How far, on the other side, should a \(60-\mathrm{kg}\) boy sit so that the seesaw will remain in balance? (A) \(0.30 \mathrm{~m}\) (B) \(0.35 \mathrm{~m}\) (C) \(0.40 \mathrm{~m}\) (D) \(0.45 \mathrm{~m}\)

Step-by-Step Solution

Answer

The boy should sit 0.45 m from the balance point.

Step 1: Understand the concept of balance on a seesaw

To keep a seesaw balanced, the moments (or torques) about the balance point must be equal for both sides. The moment is calculated by the product of the weight and the distance from the balance point. Mathematically, the moments on both sides must satisfy the equation: \[ M_1 \times d_1 = M_2 \times d_2 \] where \(M_1\) and \(M_2\) are the masses on each side, and \(d_1\) and \(d_2\) are the respective distances from the balance point.

Step 2: Identify the known variables

In this problem, we are given:Girl's mass, \(M_1 = 45 \, \text{kg}\)Girl's distance from balance point, \(d_1 = 0.6 \, \text{m}\)Boy's mass, \(M_2 = 60 \, \text{kg}\)We need to determine the boy's distance from the balance point, \(d_2\).

Step 3: Set up the balance equation

Using the balance equation \( M_1 \times d_1 = M_2 \times d_2 \), substituting the known values:\[ 45 \, \text{kg} \times 0.6 \, \text{m} = 60 \, \text{kg} \times d_2 \]

Step 4: Solve for the unknown distance

To find \(d_2\), rearrange the equation:\[ d_2 = \frac{45 \, \text{kg} \times 0.6 \, \text{m}}{60 \, \text{kg}} \]Calculate:\[ d_2 = \frac{27 \, \text{kg} \times \text{m}}{60 \, \text{kg}} \]\[ d_2 = 0.45 \, \text{m} \]

Key Concepts

torque calculation

In physics, torque refers to the rotational equivalent of force. It is sometimes called the 'moment of force'. Torque affects how an object rotates around a specific point or axis. The calculation of torque involves two key factors: force and the lever arm (distance from the pivot point).

The formula for torque (\tau) is given by: \[ \tau = F \times d \]

Here,

• \( F \) is the force applied
• \( d \) is the distance from the pivot point (lever arm).

On a seesaw, the weight of a person is the force, and the distance from the balance point to where the person is sitting is the lever arm. To balance the seesaw, the torques on either side must be equal.
For example: \( \text{Torque}_\text{girl} = \text{Torque}_\text{boy} \).
We use the equation: \[ M_1 \times d_1 = M_2 \times d_2 \]
where \( M_1 \) and \( M_2 \) are the masses on each side and \( d_1 \), \( d_2 \) are the respective distances.

moment of force

The moment of force, often referred to simply as 'moment' or 'torque', is a measure of the turning effect produced by a force. In the seesaw problem, the weight of each person creates a moment of force about the pivot point. This can be visualized as each person trying to rotate the seesaw around its fulcrum.

To determine the moment of force: \[ \text{Moment} = \text{Force} \times \text{Distance} \] Here, force is the person's weight, and the distance is how far they are from the fulcrum.

In our given problem:

• \(\text{Moment}_\text{girl} = 45 \times 0.6 = 27 \text{ kg}\text{m}\)
• \(\text{Moment}_\text{boy} = 60 \times d_2\)

We then solve for \( d_2 \) to balance these moments:
\[ d_2 = \frac{27}{60} = 0.45 \text{ meters} \] The boy must sit 0.45 meters from the fulcrum to achieve equilibrium.

equilibrium

Equilibrium refers to a state where all forces and moments acting on a system are balanced, resulting in no net motion. For a seesaw to be in equilibrium, the moments on both sides of the fulcrum must be equal. This condition prevents the seesaw from tipping to one side.

In practical terms, this means:

• \( \text{Total Moment Left} = \text{Total Moment Right} \)
• The seesaw remains level, with no rotation.

In our exercise, to find equilibrium: \[ 45 \times 0.6 = 60 \times d_2 \] Solving this gives us: \[ d_2 = 0.45 \text{ meters} \] Thus, placing the boy 0.45 meters away from the fulcrum balances the seesaw, maintaining equilibrium and ensuring it remains level.