Pulling a train. A locomotive exerts a force \(F=20,000 \mathrm{~N}\) on a train cart loaded with automobiles. The mass of the cart, including its load, is \(10,000 \mathrm{~kg}\). (a) What is the acceleration of the train cart? Another car with mass \(2000 \mathrm{~kg}\) is added to the load on the train cart. (b) What is the acceleration of the train cart now?

Newton's Second Law is key to understanding how force and acceleration work together. It tells us that the force applied to an object is equal to the mass of the object times its acceleration. This can be summarized by the formula:

• \[ F = ma \]
  

In simpler terms, if you push on something, how much it will speed up depends on how hard you push and how heavy it is. More force means more acceleration, and a heavier mass means less acceleration for the same amount of force.
In our exercise, the locomotive exerts a force of 20,000 N on the train cart. The mass of the cart is 10,000 kg. Using the formula, we can solve for acceleration:

• \[ a = \frac{F}{m} = \frac{20,000 \text{~N}}{10,000 \text{~kg}} = 2 \text{~m/s}^2 \]
  

So, the train cart accelerates at 2 m/s^2.
Now, if an additional car with mass 2,000 kg is added, the total mass becomes 12,000 kg. We use the same formula to find the new acceleration:

• \[ a = \frac{20,000 \text{~N}}{12,000 \text{~kg}} = 1.67 \text{~m/s}^2 \]
  

The acceleration decreases because the mass has increased, but the force remains the same.

Mass and weight are closely related but are not the same thing. Mass refers to the amount of matter in an object and is measured in kilograms (kg). It stays the same no matter where you are. For example, a bag of rice will always have the same mass whether you're on Earth or the Moon.
Weight, on the other hand, is the force exerted by gravity on that mass. It depends on both the mass and the gravitational pull of the place you are in. Weight is calculated by the formula:

• \[ \text{Weight} = \text{mass} \times g \]
  

where \[ g \] is the acceleration due to gravity (approximately 9.81 \[ \text{m/s}^2 \] on Earth).
For example, if our train cart were on Earth, its weight would be:

• \[ \text{Weight} = 10,000 \text{~kg} \times 9.81 \text{~m/s}^2 = 98,100 \text{~N} \]
  

Adding another 2,000 kg car to the load would increase the weight:

• \[ \text{Weight} = 12,000 \text{~kg} \times 9.81 \text{~m/s}^2 = 117,720 \text{~N} \]
  

So, mass is the same, but weight changes based on the gravitational pull of your location.

Mechanical physics deals with the behavior of physical bodies when subjected to forces or displacements. In the context of our exercise, it helps explain how the train cart moves when a force is applied.
Key concepts in mechanical physics include:

• Force: A push or pull on an object.
• Acceleration: The rate at which an object's velocity changes over time.
• Mass: The amount of matter in an object.

Newton's Second Law, which we've been using, is fundamental in this field. It states that the force exerted on an object is the product of its mass and acceleration:

• \[ F = ma \]
  

Beyond our example, mechanical physics covers many other areas like:

• Momentum: The quantity of motion an object has, dependent on its mass and velocity.
• Energy: The capacity to do work, present in various forms such as kinetic and potential energy.
• Work: The transfer of energy when an object is moved over a distance by an external force.

In our exercise, understanding the relationship between force, mass, and acceleration is crucial for predicting how the train cart will move when an additional load is added.