Find Coordinates Of Centre Of Mass Of System Given That 3 Points M1 =2 Kg M2= 1 Kg M 3=3 Kg Are Placed Respectively At Points (4 ,2) (6 ,2) And (-4 ,- 8) Coordinates Are Measured In Metre​

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Introduction

In physics, the centre of mass of a system is a point that represents the average location of the total mass of the system. It is an essential concept in understanding the motion of objects and systems. In this article, we will discuss how to find the coordinates of the centre of mass of a system given the masses and coordinates of three points.

What is Centre of Mass?

The centre of mass of a system is a point that is determined by the distribution of mass within the system. It is the point where the entire mass of the system can be considered to be concentrated for the purpose of analyzing its motion. The centre of mass is a fixed point that does not move even if the individual objects within the system are moving.

Calculating the Centre of Mass

To calculate the centre of mass of a system, we need to know the masses and coordinates of all the objects within the system. The formula for calculating the centre of mass is given by:

x = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn)

y = (m1y1 + m2y2 + ... + mnyn) / (m1 + m2 + ... + mn)

where (x, y) are the coordinates of the centre of mass, m1, m2, ..., mn are the masses of the objects, and (x1, y1), (x2, y2), ..., (xn, yn) are the coordinates of the objects.

Example Problem

Let's consider a system of three objects with masses M1 = 2 kg, M2 = 1 kg, and M3 = 3 kg. The coordinates of the objects are given as (4, 2), (6, 2), and (-4, -8) respectively. We need to find the coordinates of the centre of mass of the system.

Step 1: Calculate the x-coordinate of the centre of mass

To calculate the x-coordinate of the centre of mass, we need to multiply the masses of the objects by their respective x-coordinates and add them up.

m1x1 = 2 kg x 4 m = 8 kg m m2x2 = 1 kg x 6 m = 6 kg m m3x3 = 3 kg x (-4 m) = -12 kg m

Now, we add up the products:

8 kg m + 6 kg m - 12 kg m = 2 kg m

Step 2: Calculate the y-coordinate of the centre of mass

To calculate the y-coordinate of the centre of mass, we need to multiply the masses of the objects by their respective y-coordinates and add them up.

m1y1 = 2 kg x 2 m = 4 kg m m2y2 = 1 kg x 2 m = 2 kg m m3y3 = 3 kg x (-8 m) = -24 kg m

Now, we add up the products:

4 kg m + 2 kg m - 24 kg m = -18 kg m

Step 3: Calculate the total mass of the system

To calculate the total mass of the system, we add up the masses of the objects.

m1 + m2 + m3 = 2 kg + 1 kg + 3 kg = 6 kg

Step 4: Calculate the coordinates of the centre of mass

Now that we have the x and y coordinates of the centre of mass, we can calculate the coordinates of the centre of mass by dividing the x and y coordinates by the total mass of the system.

x = 2 kg m / 6 kg = 0.33 m y = -18 kg m / 6 kg = -3 m

Therefore, the coordinates of the centre of mass of the system are (0.33 m, -3 m).

Conclusion

In this article, we discussed how to find the coordinates of the centre of mass of a system given the masses and coordinates of three points. We used the formula for calculating the centre of mass and applied it to an example problem. The coordinates of the centre of mass are determined by the distribution of mass within the system, and they represent the average location of the total mass of the system.

References

  • Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
  • Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.

Further Reading

  • Centre of mass on Wikipedia
  • Centre of mass on Physics Classroom
  • Centre of mass on HyperPhysics
    Centre of Mass Q&A =====================

Frequently Asked Questions about Centre of Mass

In this article, we will answer some of the most frequently asked questions about centre of mass. Whether you are a student, teacher, or just someone interested in physics, this article will provide you with a comprehensive understanding of centre of mass.

Q: What is centre of mass?

A: Centre of mass is a point that represents the average location of the total mass of a system. It is an essential concept in understanding the motion of objects and systems.

Q: How is centre of mass calculated?

A: Centre of mass is calculated using the formula:

x = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn)

y = (m1y1 + m2y2 + ... + mnyn) / (m1 + m2 + ... + mn)

where (x, y) are the coordinates of the centre of mass, m1, m2, ..., mn are the masses of the objects, and (x1, y1), (x2, y2), ..., (xn, yn) are the coordinates of the objects.

Q: What is the difference between centre of mass and centre of gravity?

A: Centre of mass and centre of gravity are often used interchangeably, but they are not exactly the same thing. Centre of mass is a point that represents the average location of the total mass of a system, while centre of gravity is a point that represents the point where the weight of an object can be considered to act.

Q: Can centre of mass be outside the object?

A: Yes, centre of mass can be outside the object. This is because centre of mass is a point that represents the average location of the total mass of a system, and it can be located anywhere within the system.

Q: How does centre of mass change when objects are moving?

A: Centre of mass changes when objects are moving. The centre of mass of a system will move in the same direction and at the same speed as the centre of mass of the individual objects.

Q: Can centre of mass be used to predict the motion of objects?

A: Yes, centre of mass can be used to predict the motion of objects. By knowing the centre of mass of a system, you can predict the motion of the system as a whole.

Q: What are some real-world applications of centre of mass?

A: Centre of mass has many real-world applications, including:

  • Designing buildings and bridges
  • Understanding the motion of vehicles
  • Predicting the motion of celestial bodies
  • Understanding the behavior of complex systems

Q: How can I learn more about centre of mass?

A: There are many resources available to learn more about centre of mass, including:

  • Textbooks on physics and engineering
  • Online courses and tutorials
  • Research papers and articles
  • Professional organizations and conferences

Conclusion

In this article, we have answered some of the most frequently asked questions about centre of mass. Whether you are a student, teacher, or just someone interested in physics, this article has provided you with a comprehensive understanding of centre of mass.

References

  • Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
  • Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.
  • Centre of mass on Wikipedia
  • Centre of mass on Physics Classroom
  • Centre of mass on HyperPhysics

Further Reading

  • Centre of mass in engineering
  • Centre of mass in physics education
  • Centre of mass in research and development