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Chapter 1: Problem 56

Two persons \(\mathrm{A}\) and \(\mathrm{B}\) of weight \(80 \mathrm{~kg}\) and \(50 \mathrm{~kg}\) respectivel are standing at opposite ends of a boat of mass \(70 \mathrm{~kg}\) a length \(10 \mathrm{~m}\) at rest. When they interchange their position then displacement of the boat will be (1) \(1.5 \mathrm{~m}\) towards right (2) \(1.0 \mathrm{~m}\) towards right (3) \(1.5 \mathrm{~m}\) towards left (4) stationary

Short Answer

Expert verified
The boat will move 1.5 m towards the right.

Step by step solution

01

Determine the Initial Center of Mass

Use the center of mass formula for a system of three objects: two people and the boat. Initially, person A is at -5 meters, person B is at 5 meters, and the boat's mass is concentrated at the center (0 meters).The center of mass is calculated as follows:\[\text{Initial CM} = \frac{80(-5) + 70(0) + 50(5)}{80 + 70 + 50}\]\[= \frac{-400 + 0 + 250}{200}\]\[= \frac{-150}{200} = -0.75\, \text{m}\, \text{(towards person A from the boat center)}\].
02

Determine the New Center of Mass

After A and B interchange their positions, re-calculate the center of mass using the same formula. Now, person A is at 5 meters and person B is at -5 meters.\[\text{New CM} = \frac{80(5) + 70(0) + 50(-5)}{80 + 70 + 50}\]\[= \frac{400 + 0 - 250}{200}\]\[= \frac{150}{200} = 0.75\, \text{m}\, \text{(towards person B from the boat center)}\].
03

Calculate the Displacement of the Boat

The overall system is initially at rest and remains so. Hence, the center of mass should not actually change its location in space.To balance this shift in the human CM (from -0.75 to 0.75), the boat itself must shift in the opposite direction.Therefore, the boat must shift by a total of \(2 \times 0.75 = 1.5\, \text{m}\) towards the right.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Displacement
Displacement is all about how far something moves from its starting position, regardless of the path taken. It's important to distinguish displacement from distance. While distance considers the path traveled, displacement focuses on the shortest route -- the straight line between start and finish.
In the context of this exercise, displacement refers to the movement of the boat caused by the people changing positions. This shift happens because the center of mass has moved due to the swapping positions of the two people, A and B.
  • Displacement is directional, meaning it has both magnitude and direction.
  • In this exercise, the boat's displacement is 1.5 meters towards the right.
Displacement is a vector quantity, and understanding it helps us to measure how shifts in mass (like the two people moving) affect positions over time.
Exploring Newton's Laws
Newton's laws of motion are fundamental for understanding the physics behind moving objects. In this problem, especially, Newton鈥檚 first and third laws are at play.
  • Newton's First Law of Motion: This law, also known as the law of inertia, states that an object will remain at rest or move at a constant velocity unless acted upon by an external force. Initially, the system (boat plus people) is at rest.
  • Newton's Third Law of Motion: For every action, there is an equal and opposite reaction. When A and B switch places, each step they take exerts a force on the boat, causing it to move. The movement of one affects the other.
Using these principles, we understand that the motion of A and B swapping positions creates internal forces that lead to the boat's displacement. The boat moving right balances the internal shift of the humans' center of mass moving left.
Conservation of Momentum
The concept of momentum conservation is very useful in solving this problem. Momentum, which is the product of mass and velocity, must remain constant in a system unless acted on by an external force.
Initially, the system of the boat and people is at rest, meaning the total momentum is zero. Even after A and B have switched places, the system must continue to have zero momentum.
  • The change in position by A and B shifts their individual momentum but must not shift the system's total momentum.
  • This balance is achieved by the boat moving in the opposite direction to the shift in the center of mass caused by A and B swapping places, thereby conserving momentum overall.
By understanding these principles, students can grasp how the boat's displacement happens without any external force influencing the system鈥檚 total momentum.

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Most popular questions from this chapter

Choose the correct statements from the following: (1) The general form of Newton's second law of motion is \(\vec{F}_{\mathrm{ext}}=m \vec{a}\) (2) A body can have energy and get no momentum. (3) A body having momentum must necessarily have kinetic energy. (4) The relative velocity of two bodies in a head-on elastic collision remains unchanged in magnitude and direction.

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