Solving Physics Motion Problem Finding Force On Cubic Object

July 30, 2025 by Scholario Team 61 views

Understanding the Physics of Motion A Comprehensive Analysis

Hey guys! Let's dive into the fascinating world of physics, where we'll tackle a classic problem involving motion, force, and energy. We're going to break down a scenario where a 1 kg cubic object is pushed across a frictionless table. This is a super common type of problem in introductory physics, and understanding it will give you a solid foundation for more advanced topics. So, buckle up, and let's get started!

Imagine a 1 kg cubic object sitting perfectly still on a smooth, frictionless horizontal table. Now, a constant force comes into play, pushing this object. As a result, the object starts moving from rest and speeds up until it reaches a velocity of 3 m/s after traveling a distance of 1.5 meters. The big question we want to answer is: What is the magnitude (or size) of this force that's causing the object to move and accelerate? To solve this, we'll need to use some key concepts from physics, including Newton's laws of motion and the work-energy theorem.

Before we jump into equations, let's take a moment to really understand what's happening. We have an object initially at rest, which means its starting velocity is zero. A force acts on it, causing it to accelerate. This acceleration means the object's velocity is changing over time. We know the final velocity (3 m/s) and the distance it traveled (1.5 m). Our goal is to find the magnitude of the force. Think of it like pushing a box across a smooth floor – the harder you push (the more force you apply), the faster it will go and the quicker it will reach a certain speed.

In this context, the absence of friction is super important. Friction is a force that opposes motion, like when you try to slide something across a rough surface. Because there's no friction in our problem, all the force we apply goes directly into accelerating the object. This simplifies our calculations quite a bit! Now that we have a good grasp of the situation, let's look at the physics principles we'll use to solve it.

To solve this problem, we'll rely on two major pillars of physics:

1. Newton's Second Law of Motion:

This is the cornerstone of classical mechanics! Newton's Second Law tells us that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, we write this as F = ma, where:

F is the net force (the total force acting on the object).
m is the mass of the object.
a is the acceleration of the object.

In our case, since there's only one force acting horizontally (the one we're trying to find) and no friction, the net force is simply the applied force. So, if we can figure out the object's acceleration, we can easily calculate the force using this law.

2. Work-Energy Theorem:

This theorem provides a powerful connection between work and energy. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In simpler terms, the amount of energy you put into moving something is equal to the change in its motion. The formula for work done by a constant force is W = Fd, where:

W is the work done.
F is the force applied.
d is the distance over which the force is applied.

Kinetic energy (KE) is the energy an object possesses due to its motion. It's given by the formula KE = (1/2)mv^2, where:

KE is the kinetic energy.
m is the mass of the object.
v is the velocity of the object.

The change in kinetic energy (ΔKE) is simply the final kinetic energy minus the initial kinetic energy. In our problem, the object starts from rest, so its initial kinetic energy is zero. This simplifies things nicely!

By combining these concepts, we can relate the force, distance, mass, and velocity, allowing us to solve for the unknown force.

Alright, let's put these concepts into action and solve for the magnitude of the force. We'll break it down step-by-step to make it super clear.

1. Calculate the Change in Kinetic Energy:

First, we need to find how much the object's kinetic energy changed. Remember, the object started at rest (0 m/s) and ended up moving at 3 m/s. We'll use the kinetic energy formula: KE = (1/2)mv^2.

Initial KE = (1/2) * (1 kg) * (0 m/s)^2 = 0 Joules (J)
Final KE = (1/2) * (1 kg) * (3 m/s)^2 = (1/2) * 1 kg * 9 m^2/s2 = 4.5 J

So, the change in kinetic energy (ΔKE) is the final KE minus the initial KE: ΔKE = 4.5 J - 0 J = 4.5 J

2. Apply the Work-Energy Theorem:

The work-energy theorem tells us that the work done on the object is equal to the change in its kinetic energy. We know the change in kinetic energy (4.5 J) and the distance the object traveled (1.5 m). We also know that work (W) is equal to force (F) times distance (d): W = Fd.

So, we can set up the equation: Fd = ΔKE, which becomes F * (1.5 m) = 4.5 J

3. Solve for the Force:

Now, it's just a matter of solving for F. Divide both sides of the equation by 1.5 m: F = 4.5 J / 1.5 m = 3 Newtons (N)

And there you have it! The magnitude of the constant force acting on the cubic object is 3 Newtons. That wasn't so bad, right? We used the work-energy theorem to connect the change in kinetic energy to the force and distance, and then we simply solved for the force.

Just to show you there's often more than one way to crack a physics problem, let's tackle this using a different approach. This time, we'll use kinematics (the study of motion) to find the acceleration first, and then use Newton's Second Law to find the force.

1. Use Kinematics to Find Acceleration:

We can use one of the constant acceleration equations to relate initial velocity (v₀), final velocity (v), acceleration (a), and distance (d). The equation that fits our needs is: v^2 = v₀^2 + 2ad

We know: v = 3 m/s, v₀ = 0 m/s, and d = 1.5 m. Plug these values into the equation:

(3 m/s)^2 = (0 m/s)^2 + 2 * a * (1.5 m)

9 m^2/s2 = 0 + 3 m * a

Now, solve for a: a = (9 m^2/s2) / (3 m) = 3 m/s^2

So, the object's acceleration is 3 m/s^2. This means its velocity is increasing by 3 meters per second every second.

2. Apply Newton's Second Law:

Now that we know the acceleration, we can use Newton's Second Law (F = ma) to find the force. We know the mass (m = 1 kg) and the acceleration (a = 3 m/s^2).

F = (1 kg) * (3 m/s^2) = 3 N

Again, we arrive at the same answer: the magnitude of the force is 3 Newtons. This demonstrates how different physics principles can be used together to solve the same problem, giving you a deeper understanding of the concepts.

Awesome! We successfully determined the magnitude of the force acting on the cubic object using two different methods. First, we used the work-energy theorem, which related the work done by the force to the change in kinetic energy. Then, we used kinematics to find the acceleration and applied Newton's Second Law to calculate the force. Both approaches led to the same result: a force of 3 Newtons.

This problem is a great example of how physics principles connect and can be applied in different ways. Understanding these concepts is crucial for solving more complex problems in mechanics and other areas of physics. Remember, the key is to break down the problem, identify the relevant concepts, and apply the appropriate equations. Keep practicing, and you'll become a physics whiz in no time! Keep your curiosity sparked, and happy problem-solving!

Q1: What would happen if there was friction between the object and the table? A: If there were friction, some of the applied force would be used to overcome the frictional force, and the object's acceleration would be less. We'd need to account for the frictional force in our calculations, making the problem slightly more complex.

Q2: How would the result change if the mass of the object were different? A: If the mass were different, the force required to achieve the same acceleration would also be different, according to Newton's Second Law (F = ma). A heavier object would require a larger force to reach the same final velocity over the same distance.

Q3: Could we use conservation of energy to solve this problem? A: Yes, we could use the principle of conservation of energy. The work done by the force is converted into kinetic energy of the object. This is essentially what the work-energy theorem states, so it's another way to approach the problem.

Q4: What if the force wasn't constant? How would we solve it then? A: If the force wasn't constant, the acceleration wouldn't be constant either. We'd likely need to use calculus to solve the problem, as the equations we used here assume constant acceleration. We might need to integrate the force over the distance to find the work done.

Q5: Where can I find more problems like this to practice? A: You can find similar problems in introductory physics textbooks, online physics resources, and practice problem sets. Look for problems involving Newton's laws of motion, work, energy, and kinematics. The more you practice, the better you'll become at solving these types of problems!