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mihalych1998 [28]
2 years ago
12

Consider the vector field. f(x, y, z) = xy2z2i x2yz2j x2y2zk (a) find the curl of the vector field?

Physics
1 answer:
Marat540 [252]2 years ago
5 0

Observe that the given vector field is a gradient field:

Let f(x,y,z)=\nabla g(x,y,z), so that

\dfrac{\partial g}{\partial x} = x y^2 z^2

\dfrac{\partial g}{\partial y} = x^2 y z^2

\dfrac{\partial g}{\partial z} = x^2 y^2 z

Integrating the first equation with respect to x, we get

g(x,y,z) = \dfrac12 x^2 y^2 z^2 + h(y,z)

Differentiating this with respect to y gives

\dfrac{\partial g}{\partial y} = x^2 y z^2 + \dfrac{\partial h}{\partial y} = x^2 y z^2 \\\\ \implies \dfrac{\partial h}{\partial y} = 0 \implies h(y,z) = i(z)

Now differentiating g with respect to z gives

\dfrac{\partial g}{\partial z} = x^2 y^2 z + \dfrac{di}{dz} = x^2 y^2 z \\\\ \implies \dfrac{di}{dz} = 0 \implies i(z) = C

Putting everything together, we find a scalar potential function whose gradient is f,

f(x,y,z) = \nabla \left(\dfrac12 x^2 y^2 z^2 + C\right)

It follows that the curl of f is 0 (i.e. the zero vector).

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A 59.31 kg rock is sitting at the top of a cliff that is 300 m tall. What is the gravitational potential energy of that rock?
Diano4ka-milaya [45]

Answer:

The gravitational potential energy of that rock is 174371.4 J.

Explanation:

Given

  • Mass m = 59.31 kg
  • Height h = 300 m

To determine

We need to find the gravitational potential energy of the rock

We know that the potential energy of a body is termed as the stored energy due to its position.

One kind of energy comes from Earth's gravity — Gravitational potential energy (GPE).

Gravitational potential energy (GPE) can be determined using the formula

GPE = mgh

where

  • m is the mass
  • g is the gravitational acceleration which is equal to g = 9.8 m/s²
  • h is the height
  • GPE is the Gravitational potential energy

now substituting m = 59.31, h = 300 and g = 9.8

GPE = mgh

         =59.31\times 9.8\times 300

         =174371.4 J

Therefore, the gravitational potential energy of that rock is 174371.4 J.

4 0
3 years ago
The answer please it’s very simple this is 7th grade science
allochka39001 [22]

The greatest point for kinetic is at the bottom and in the middle it is in half and at the top it is at the highest in potential energy.

3 0
3 years ago
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Explain why, if you go to the moon, your weight would change but your mass wouldnt
Evgen [1.6K]
There's little gravity so your weight would change but not your mass
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You toss a ball in the air. what is the work done by gravity as the ball goes up?
jolli1 [7]

Answer: Gravity slows the ball down as it goes up and eventually stops it from going up and starts to pull it back down to earth.

Explanation:

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Neurons in our bodies carry weak currents that produce detectable magnetic fields. A technique called magnetoencephalography, or
Stolb23 [73]

Answer:

I = (1.80 × 10⁻¹⁰) A

Explanation:

From Biot Savart's law, the magnetic field formula is given as

B = (μ₀I)/(2πr)

B = magnetic field = (1.0 × 10⁻¹⁵) T

μ₀ = magnetic constant = (4π × 10⁻⁷) H/m

r = 3.6 cm = 0.036 m

(1.0 × 10⁻¹⁵) = (4π × 10⁻⁷ × I)/(2π × 0.036)

4π × 10⁻⁷ × I = 1.0 × 10⁻¹⁵ × 2π × 0.036

I = (1.80 × 10⁻¹⁰) A

Hope this Helps!!!

3 0
4 years ago
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