Question

3 points particles with masses 4kg,2kg,3kg are at the corners of equilateral triangle of side m =2m.

Answer 1

[AI-generated answer — Explain Bot]

To find the gravitational potential energy of the system, we can use the formula:

𝑃𝐸 = −𝐺 × 𝑚1 × 𝑚2 ÷ 𝑟

Where:
𝑃𝐸 is the gravitational potential energy
𝐺 is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2)
𝑚1 and 𝑚2 are the masses of the particles
𝑟 is the distance between the particles

Since the triangle is equilateral, all sides are of equal length. Let's consider side "m", which is 2 meters long.

Now, to calculate the gravitational potential energy between two particles, we need to find the distance between them. In an equilateral triangle, the distance between any two particles can be expressed as a fraction of the side length "m".

Let's label the particles as A, B, and C. The distance between A and B is "dAB", the distance between B and C is "dBC", and the distance between C and A is "dCA".

From the properties of an equilateral triangle, we know that the height of the triangle (from any corner to the midpoint of the opposite side) is √3/2 times the side length. Therefore, the height of the triangle is (√3/2) × 2 = √3 meters.

To calculate the distances between the particles, we can use the Pythagorean theorem. Since the height of the equilateral triangle bisects the base, we have two right-angled triangles.

In triangle ABC, with A being the top vertex and BC being the base:
- The height of the triangle (√3 meters) is the opposite side.
- Half of BC (which is m/2 = 2/2 = 1 meter) is the adjacent side.

Using the Pythagorean theorem, we can calculate the distance between particles A and B:

dAB^2 = (1 meter)^2 + (√3 meters)^2
dAB^2 = 1 meter^2 + 3 meters^2
dAB^2 = 1 + 3
dAB^2 = 4

Therefore, dAB = √4 = 2 meters.

Similarly, we can find the distances dBC and dCA.
dBC = 2 meters
dCA = 2 meters

Now that we have the distances between the particles, we can calculate the gravitational potential energy for each pair of particles:

For particles A and B:
𝑃𝐸AB = −𝐺 × 4 kg × 2 kg ÷ 2 meters

For particles B and C:
𝑃𝐸BC = −𝐺 × 2 kg × 3 kg ÷ 2 meters

For particles C and A:
𝑃𝐸CA = −𝐺 × 3 kg × 4 kg ÷ 2 meters

Finally, to find the total gravitational potential energy of the system, we sum up the potential energies between all pairs of particles:

𝑇𝑜𝑡𝑎𝑙 𝑃𝐸 = 𝑃𝐸AB + 𝑃𝐸BC + 𝑃𝐸CA

which gives:

𝑇𝑜𝑡𝑎𝑙 𝑃𝐸 = (−𝐺 × 4 kg × 2 kg ÷ 2 meters) + (−𝐺 × 2 kg × 3 kg ÷ 2 meters) + (−𝐺 × 3 kg × 4 kg ÷ 2 meters)

Simplifying this equation will give you the numerical value for the total gravitational potential energy of the system.