Question

A system consists of two particles. Particle 1 with mass \(2.00 \mathrm{~kg}\) is located at \((2.00 \mathrm{~m}, 6.00 \mathrm{~m})\) and has a velocity of \((4.00 \mathrm{~m} / \mathrm{s}, 2.00 \mathrm{~m} / \mathrm{s})\) Particle 2 with mass \(3.00 \mathrm{~kg}\) is located at \((4.00 \mathrm{~m}, 1.00 \mathrm{~m})\) and has a velocity of \((0,4.00 \mathrm{~m} / \mathrm{s})\) a) Determine the position and the velocity of the center of mass of the system. b) Sketch the position and velocity vectors for the individual particles and for the center of mass.

Short answer

Question: Calculate the position and velocity of the center of mass for a two-particle system with the following information: - Particle 1: mass \(m_1 = 2.00 kg\), position vector \(\vec{r}_1 = (2.00 m, 6.00 m)\), velocity vector \(\vec{v}_1 = (4.00 m/s, 2.00 m/s)\) - Particle 2: mass \(m_2 = 3.00 kg\), position vector \(\vec{r}_2 = (4.00 m, 1.00 m)\), velocity vector \(\vec{v}_2 = (0, 4.00 m/s)\) Additionally, sketch the position and velocity vectors for the individual particles and the center of mass on a coordinate plane.

Step-by-step solution

Calculate the position of the center of mass

In order to find the position of the center of mass, we need to use the equation: $$\vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$$ We have the following mass and position vectors for both particles: - Particle 1: mass \(m_1 = 2.00 kg\), position vector \(\vec{r}_1= (2.00 m, 6.00 m)\) - Particle 2: mass \(m_2 = 3.00 kg\), position vector \(\vec{r}_2 = (4.00 m, 1.00 m)\) Now, let's calculate the center of mass position vector: $$\vec{R}_{cm} = \frac{(2.00 kg)(2.00 m, 6.00 m) + (3.00 kg)(4.00 m, 1.00 m)}{2.00 kg + 3.00 kg}$$

Calculate the velocity of the center of mass

Now, we need to find the velocity of the center of mass. We can do this by using the equation: $$\vec{V}_{cm} = \frac{\sum m_i \vec{v}_i}{\sum m_i}$$ We have the following mass and velocity vectors for both particles: - Particle 1: mass \(m_1 = 2.00 kg\), velocity vector \(\vec{v}_1 = (4.00 m/s, 2.00 m/s)\) - Particle 2: mass \(m_2 = 3.00 kg\), velocity vector \(\vec{v}_2= (0, 4.00 m/s)\) Now, let's calculate the center of mass velocity vector: $$\vec{V}_{cm} = \frac{(2.00 kg)(4.00 m/s, 2.00 m/s) + (3.00 kg)(0, 4.00 m/s)}{2.00 kg + 3.00 kg}$$

Sketch position and velocity vectors

For this step, we will sketch the position and velocity vectors for each individual particle and the center of mass. 1. Draw a coordinate plane (x-y axis). 2. Draw particle 1 position vector (\(\vec{r}_1\)) with origin at \((0,0)\) and tip at \((2.00 m, 6.00 m)\). 3. Draw particle 2 position vector (\(\vec{r}_2\)) with origin at \((0,0)\) and tip at \((4.00 m, 1.00 m)\). 4. Calculate and draw the center of mass position vector (\(\vec{R}_{cm}\)) with origin at \((0,0)\) and tip at the calculated position. 5. At each particle position vector tip, draw the individual velocity vectors (\(\vec{v}_1\) and \(\vec{v}_2\)) with tip pointing in the direction and magnitude of the velocities. 6. At the center of mass position vector tip, draw the center of mass velocity vector (\(\vec{V}_{cm}\)) with tip pointing in the direction and magnitude of the calculated center of mass velocity. The resulting sketch should have position and velocity vectors for both particles and the center of mass. With the graph, we can visualize how the individual particles and the center of mass move in the system.

Key concepts

System of Particles

When we refer to a 'system of particles,' we are considering multiple particles as a single entity to study their collective behavior. This concept simplifies analysis in physics, especially when we look at how forces affect the motion and interactions of particles. In the provided exercise, the system consists of two particles, each with distinct masses, positions, and velocities. The first step in understanding such a system is by determining the center of mass, which is the point where the system’s mass is concentrated and can be thought of as the 'average' position of all the particles' mass.

The calculation of the center of mass involves combining the masses and position vectors of the individual particles. Importantly, the center of mass can move even if the system as a whole doesn’t translate, as it depends on relative positions of the particles making up the system. The concept is crucial as it helps us predict the motion of the system under the influence of external forces.

Velocity Vectors

Velocity vectors in physics represent both the speed and direction of an object’s motion. They are visualized as arrows pointing in the direction of travel; the length of the arrow is proportional to the magnitude or speed. In our exercise, each particle has its velocity vector defined, which helps in determining the collective motion of the system through the calculation of the center of mass velocity.

When dealing with a system of particles, the velocity vectors play a pivotal role in finding the velocity of the center of mass. This is computed as a mass-weighted average of the individual velocity vectors. Understanding how to represent and calculate these vectors is fundamental for students since it enables the analysis of the kinetic aspects of systems, from the movement of celestial bodies to the dynamics of molecules.

Position Vectors

Position vectors, on the other hand, denote the location of a particle relative to an origin point in a coordinate system. These vectors are essential for denoting where an object is in space and serve as a starting point for calculating displacement, velocity, and acceleration.

In the context of our exercise, position vectors are used to determine the location of each particle, which subsequently aids in calculating the position of the system’s center of mass. Each particle’s position vector represents the distance and direction from the origin to the particle’s location. By understanding position vectors, students can graphically depict how particles in a system are arranged in space and will have a better grasp of the concepts related to the geometry of motion.