Problem 103 - Algebra and Trigonometry

Question

Explaining the Concepts Use vectors to explain why it is difficult to hold a heavy stack of books perfectly still for a long period of time. As you become exhausted, what eventually happens? What does this mean in terms of the forces acting on the books?

Step-by-Step Solution

Answer

Initially two equal and opposite forces - the force of gravity and the force applied by the person - keep the stack of books stationary. As the person becomes exhausted, the force they can apply reduces, becoming less than the force of gravity. This imbalance in forces causes the stack of books to start moving downwards.

Step 1: Understanding and Identifying the Forces Involved

Identify the forces at play when holding a stack of books. Two crucial forces are involved: gravitational force (\( \vec{F}_g \)) and the force exerted by the person (\( \vec{F}_p \)). Initially, when the person is not exhausted, \( \vec{F}_p \) is equal and opposite to \( \vec{F}_g \), which keeps the books stationary.

Step 2: Interpret the Impact of Exhaustion

There is less force exerted by the person as fatigue sets in. This means \( \vec{F}_p \) becomes less than \( \vec{F}_g \).

Step 3: Explain the Outcome in Terms of Forces

The net force acting on the stack of books becomes \( \vec{F}_g - \vec{F}_p \), since \( \vec{F}_g > \vec{F}_p \). This positive net force in the direction of gravity means the stack of books will start to move downwards.

Key Concepts

Gravitational Force

Imagine holding an apple in your hand. It feels heavy, right? That's because of a force called gravity, which pulls the apple toward the center of the Earth. Gravity acts on every object with mass, which means it affects that heavy stack of books you're trying to hold up as well.

Gravity is a fundamental force in physics, represented by a vector we call the gravitational force, or \( \vec{F}_g \). It points straight down towards the Earth's center and is inescapable; whether you're standing on the ground or holding something up, this force is always at work. The strength of the gravitational force is determined by the mass of the books and the acceleration due to gravity, which is approximately \( 9.81m/s^2 \) on the surface of the Earth. So, \( \vec{F}_g \) for the stack of books is a constant value that can be calculated by multiplying the mass of the books by this acceleration.

Force Exerted by a Person

Have you ever wondered why holding a backpack feels lighter at the beginning of the day than toward the end? This has to do with the force you're exerting to counteract gravity. When you're holding a stack of books, you're applying a force \( \vec{F}_p \) that is ideally equal but opposite to the gravitational force. This applied force is another vector that points upward as you're trying to prevent the books from falling.

The amount of force a person can exert depends on their strength and stamina. As fatigue creeps in, the muscles generate less force and \( \vec{F}_p \) starts to decrease. This decline isn't immediate; it happens gradually as energy stores are depleted. Hence, there's a transition period during which the force exerted by the person changes from being equal to gravitational force to becoming less and less, indicating the person is getting exhausted.

Net Force

Think about when you push a stalled car; it only moves if the push is strong enough to overcome its inertia. Analogously, the net force \( \vec{F}_{net} \) on the stack of books is the sum of all the forces acting on it. Initially, when \( \vec{F}_p = -\vec{F}_g \) and they balance each other, \( \vec{F}_{net} \) is zero, meaning there's no movement.

As your muscles begin to tire, \( \vec{F}_p \) reduces, and since \( \vec{F}_g \) is relentless and unchanged, the net force becomes nonzero. In a nutshell, \( \vec{F}_{net} = \vec{F}_g - \vec{F}_p \) now has a value in the direction of gravity. The books will start to accelerate downwards, slipping from your grip, as dictated by Newton's second law of motion. It's the unbalanced net force that eventually dictates the motion of the books as your strength wanes.