4.4. NEWTON’S SECOND LAW OF MOTION

The first law refers to the simple case when the net external force on a body is zero. The second law of motion refers to the general situation when there is a net external force acting on the body. It relates the net external force to the acceleration of the body.

Newton's Second Law builds on the First Law. While the First Law applies when there is no net force acting on an object (indicating that it remains at rest or in uniform motion), the Second Law describes what happens when a net force does exist. Specifically, it states that the acceleration of an object is directly proportional to the net external force acting on it and inversely proportional to its mass. In mathematical terms, this relationship is expressed as F = ma, where F is the force, m is the mass, and a is the acceleration.

Imagine pushing a shopping cart at the grocery store. If you push gently (a small force), the cart moves slowly (less acceleration). However, if you push harder (a larger force), the cart accelerates more quickly. The heavier the cart (more mass) the harder you need to push it to achieve the same acceleration.

Momentum of a body is defined to be the product of its mass m and velocity v, and is denoted by p: p = m v.

Momentum is a key concept in understanding movement and force. It is defined as the product of an object's mass and its velocity, making it a vector quantity, which means it has both magnitude and direction. The greater the mass or the velocity of an object, the larger its momentum. This helps explain why a heavy truck moving at a moderate speed has more momentum than a small bicycle moving fast. In practical terms, momentum shows how difficult it would be to stop an object.

Consider playing a game of football. Kicking a heavy ball (large mass) will require more effort to stop it than a light beach ball, even if both are kicked at the same speed. This is because the heavy ball has much more momentum.

The greater the change in the momentum in a given time, the greater is the force that needs to be applied.

This point emphasizes the relationship between force, momentum, and time. If a large force is applied over a short period, it can cause a significant change in momentum. Conversely, applying a smaller force over a longer time can also result in the same change in momentum. This means that how quickly a force acts is crucial in determining its impact. Thus, in contexts like sports or vehicle braking, understanding how quickly you need to apply a force can help achieve the desired effect.

Think about catching a fast-moving cricket ball. A seasoned cricketer will pull their hands back while catching to extend the time over which they stop the ball, thus applying less force and reducing the chance of injury. In contrast, a novice may try to catch it with stiff hands, applying a sudden large force that can cause pain.

The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts.

This formulation of Newton's Second Law ties back to momentum. Essentially, when a force is applied to an object, it changes the object's momentum over time. The faster the change occurs, the greater the force necessary. This law can also be expressed mathematically, emphasizing that the change in momentum (Δp) over time (Δt) is equal to the force (F) applied, leading to the equation F = dp/dt. This illustrates that force is not only about how much but also how quickly it is applied.

Think of a scenario where you're playing basketball. When you push the ball away quickly, you must apply a strong force to alter its momentum quickly. Conversely, if you gently roll the ball down the court (a slower change of momentum), a lighter touch is all that’s needed.

The unit of force has not been defined so far. In fact, we use Eq. (4.4) to define the unit of force. We, therefore, have the liberty to choose any constant value for k. For simplicity, we choose k = 1. The second law then is F = ma.

The SI unit of force is derived from the relationship established in the Second Law. By setting the constant of proportionality (k) to one, we can simplify calculations, ultimately defining the newton (N) as the force that causes a mass of one kilogram to accelerate at one meter per second squared (1 N = 1 kg·m/s²). This is foundational because it lays the groundwork for understanding how forces interact with different masses.

Consider lifting a backpack. If the backpack is heavy, you can feel you need to exert a greater force to lift it than a light bag. Here, your exertion is effectively translating the mass of the bag into the required force calculated in newtons.

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