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Chapter 4: Problem 2

Two people in a rowboat wish to move around without causing the boat to move. What should be true about their total momentum? Explain.

Short Answer

Expert verified
Their total momentum should remain zero for the boat to stay still.

Step by step solution

01

Understand Momentum

Momentum is the product of an object's mass and velocity, given by the formula \( p = mv \), where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity.
02

Consider the System

Think about the system consisting of the two people and the rowboat. For the boat to remain stationary, the total momentum of this system should remain zero.
03

Apply Conservation of Momentum

According to the law of conservation of momentum, in the absence of external forces, the total momentum of a closed system should remain constant. For the rowboat and its occupants, this means their combined momentum before and after their movements should be zero.
04

Analyze the Required Condition

If both people inside the rowboat move such that their individual momenta cancel each other out, the total momentum remains zero. This implies that one person's mass and velocity should create momentum equal in magnitude but opposite in direction to the other's momentum.

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Key Concepts

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

Momentum
Momentum is a fundamental property of moving objects, defined as the product of an object's mass and velocity. Ruled by the equation \( p = mv \), where \( p \) represents momentum, \( m \) denotes mass, and \( v \) signifies velocity, it offers a measure of how much motion an object possesses. More mass or higher speed results in greater momentum. In practical terms, for two people in a rowboat to move without causing the boat itself to drift, their total combined momentum must be carefully considered. By ensuring their individual movements generate equal but opposite momenta, they maintain a neutral impact on the boat's movement.
Overall, understanding momentum helps in predicting and controlling the motion of objects in everyday situations.
Closed System
A closed system refers to a collection of objects or entities that do not interact with external influences. In the context of physics, understanding this concept helps in analyzing different scenarios, especially those involving conservation laws. When two people are in a rowboat, together with the boat itself, they form a closed system if they are isolated from external forces like water currents or wind. In this scenario, the total momentum within the system is conserved. Hence, for the rowboat to stay put while people move inside it, their interactions must not result in any net changes to the system's momentum.
This means any movement within the system should comply with conservation laws, preserving equilibrium.
External Forces
External forces are influences from outside a defined system that can alter the motion and state of objects within that system. In everyday experiences, external forces include factors such as gravity, friction, or wind. In the case of the people and their rowboat, should they be free from such external interferences, their internal movements alone dictate the system's momentum. For the boat to remain stationary, it is crucial that there are no external forces acting on it鈥攅nsuring that it behaves as a true closed system. This allows the conservation of momentum to properly apply.
Any introduction of external forces, like a sudden gust of wind, would disrupt the balance, causing the boat to shift. Therefore, to achieve the solution provided in the exercise, such influences must be controlled or eliminated.

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Most popular questions from this chapter

Two wheels of radius \(r\) rotate in the same vertical plane with angular velocities \(+\Omega\) and \(-\Omega\) about axes that are parallel and at the same height. The wheels touch one another at a point on their circumferences, so that their rotations mesh like gears in a gear train. A board is laid on top of the wheels, so that two friction forces act upon it, one from each wheel. Characterize the three qualitatively different types of motion that the board can exhibit, depending on the initial conditions.

A person on a bicycle is to coast down a ramp of height \(h\) and then pass through a circular loop of radius \(r\). What is the smallest value of \(h\) for which the cyclist will complete the loop without falling? (Ignore the kinetic energy of the spinning wheels.)(answer check available at lightandmatter.com)

(solution in the pdf version of the book) In each case, identify the force that causes the acceleration, and give its Newton'sthird-law partner. Describe the effect of the partner force. (a) A swimmer speeds up. (b) A golfer hits the ball off of the tee. (c) An archer fires an arrow. (d) A locomotive slows down.

A flexible rope of mass \(m\) and length \(L\) slides without friction over the edge of a table. Let \(x\) be the length of the rope that is hanging over the edge at a given moment in time. (a) Show that \(x\) satisfies the equation of motion \(d^{2} x / d t^{2}=g x / L .\) [Hint: Use \(F=d p / d t\), which allows you to handle the two parts of the rope separately even though mass is moving out of one part and into the other.] (b) Give a physical explanation for the fact that a larger value of \(x\) on the right-hand side of the equation leads to a greater value of the acceleration on the left side. (c) When we take the second derivative of the function \(x(t)\) we are supposed to get essentially the same function back again, except for a constant out in front. The function \(e^{x}\) has the property that it is unchanged by differentiation, so it is reasonable to look for solutions to this problem that are of the form \(x=b e^{c t}\), where \(b\) and \(c\) are constants. Show that this does indeed provide a solution for two specific values of \(c\) (and for any value of \(b\) ). (d) Show that the sum of any two solutions to the equation of motion is also a solution. (e) Find the solution for the case where the rope starts at rest at \(t=0\) with some nonzero value of \(x\).

(solution in the pdf version of the book) When the contents of a refrigerator cool down, the changed molecular speeds imply changes in both momentum and energy. Why, then, does a fridge transfer power through its radiator coils, but not force?
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