Problem 1 - Calculus Early Transcendentals

Question

Explain how to find the balance point for two people on opposite ends of a (massless) plank that rests on a pivot.

Step-by-Step Solution

Answer

Question: Two people of different masses are standing on opposite ends of a massless plank resting on a pivot. Calculate the position of the pivot to balance the plank. Answer: To find the position of the pivot relative to person 1 (d_1), use the derived equation: d_1 = (m_2 × d_2) / m_1, where m_1 and m_2 are the masses of the two people, and d_2 is the distance between person 2 and the pivot. This balance point ensures that the net moment about the pivot is zero, allowing the plank to remain balanced and not rotate.

Step 1: Understand the problem and identify the given information

In this problem, we are given: 1. A massless plank. 2. Two people on opposite ends of the plank. 3. The plank is rested on a pivot. Our task is to find the position of the pivot to balance the plank.

Step 2: Set up the principle of moments equation

According to the principle of moments, the sum of the moments or torques about the pivot must be zero for the plank to be in equilibrium. The moment is the product of the force and the perpendicular distance between the force's line of action and the pivot point. The equation for the principle of moments is: Moment_About_Pivot = Weight_Person1 × Distance_Person1_To_Pivot - Weight_Person2 × Distance_Person2_To_Pivot = 0

Step 3: Define the given variables

Let's define the given variables: Weight_Person1 = m_1 × g (where m_1 is the mass of person 1 and g is the gravitational acceleration) Weight_Person2 = m_2 × g (where m_2 is the mass of person 2) Distance_Person1_To_Pivot = d_1 Distance_Person2_To_Pivot = d_2 Now the equation becomes: Moment_About_Pivot = m_1 × g × d_1 - m_2 × g × d_2 = 0

Step 4: Solve the equation for the desired variable

In this case, we want to find the required position of the pivot, so we need to express either d_1 or d_2 in terms of the other. For this problem, let's find d_1 in terms of d_2: m_1 × g × d_1 = m_2 × g × d_2 Now divide both sides of the equation by g: m_1 × d_1 = m_2 × d_2 Now, solve the equation for d_1: d_1 = (m_2 × d_2) / m_1 The position of the pivot relative to person 1 (d_1) is now expressed in terms of the distance between person 2 and the pivot (d_2) and the masses of the two people.

Step 5: Apply the result

To find the balance point for the two people on the plank, you can now use the derived equation: d_1 = (m_2 × d_2) / m_1 With the masses of the two people and the distance between person 2 and the pivot, you can now calculate the necessary position of the pivot (d_1) relative to person 1 for the plank to be balanced. This balance point ensures that the net moment about the pivot is zero, and the plank will not rotate.