Solution to problem 14.5.7 from the collection of Kepe O.E.

14.5.7 A material point with mass m = 1 kg moves according to the law: x = 2t, y = t3, z = t4.

It is necessary to determine the angular momentum of this point relative to the Oy axis at the time t = 2 s.

(Answer -96)

To solve the problem, it is necessary to find the speed of the material point at the moment of time t=2s. To do this, it is necessary to differentiate the expressions for coordinates in time:

• v_x = 2 m/s
• v_y = 3t² m/s
• v_z = 4t³ m/s

Then, using the formula for angular momentum about an axis that passes through the origin and is parallel to the Oy axis:

M_Oy = ∫(xv_z - zv_x)dt

we substitute the found values ​​of coordinates and velocities, as well as the limits of integration (from 0 to 2 s):

M_Oy = ∫₀²(2t * 4t³ - t⁴ * 2)dt = -96 N * m * sec

Thus, the angular momentum of this point relative to the Oy axis at time t=2s is equal to -96 N * m * sec.

Solution to problem 14.5.7 from the collection of Kepe O..

We present to your attention the solution to problem 14.5.7 from the collection of Kepe O.. in electronic format. This digital product is an ideal choice for students and teachers who study physics.

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We present to you the solution to problem 14.5.7 from the collection of Kepe O.?. in electronic format. This product is a digital ideal choice for students and teachers who study physics.

To solve the problem, it is necessary to find the speed of the material point at the moment of time t=2s. To do this, it is necessary to differentiate the expressions for coordinates in time:

vx = 2 m/s vy = 3t^2 m/s vz = 4t^3 m/s

Then, using the formula for angular momentum about an axis that passes through the origin and is parallel to the Oy axis:

MOy = ∫(xvz - zvx)dt

we substitute the found values ​​of coordinates and velocities, as well as the limits of integration (from 0 to 2 s):

MOy = ∫0^2(2t * 4t^3 - t^4 * 2)dt = -96 N * m * sec

Thus, the angular momentum of this point relative to the Oy axis at time t=2s is equal to -96 N * m * sec.

Solving the problem includes a detailed analysis and step-by-step solution using the necessary formulas and methods. Beautiful design allows you to easily and conveniently familiarize yourself with the solution to the problem on any device and anywhere in the world.

By receiving this digital product, you will have access to a high-quality solution to a problem that will help you better understand physical processes and learn how to solve similar problems yourself. Don't miss the opportunity to purchase this product and improve your knowledge of physics!

The digital product is a solution to problem 14.5.7 from the collection of Kepe O.?. in electronic format. This product is intended for students and teachers involved in physics who want to improve their knowledge in this area.

The problem gives the motion of a material point with a mass of 1 kg according to the law: x = 2t, y = t3, z = t4. To determine the angular momentum relative to the Oy axis at the time t = 2 s, it is necessary to find the speed of the material point at a given time. To do this, you need to differentiate the expressions for coordinates in time: vx = 2 m/s, vy = 3t2 m/s, vz = 4t3 m/s.

Then, using the formula for the angular momentum about an axis that passes through the origin and is parallel to the Oy axis: MOy = ∫(xvz - zvx)dt, we substitute the found values ​​of coordinates and velocities, as well as the limits of integration (from 0 to 2 s): MOy = ∫0^2 (2t * 4t3 - t4 * 2)dt = -96 N * m * sec.

Thus, the solution to problem 14.5.7 from the collection of Kepe O.?. includes detailed analysis and step-by-step solution using necessary formulas and methods. Beautiful html design allows you to easily and conveniently familiarize yourself with the solution to the problem on any device and anywhere in the world. Purchasing this digital product will allow you to improve your knowledge of physics and learn to solve similar problems yourself.

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Problem 14.5.7 from the collection of Kepe O.?. consists in determining the angular momentum of a material point weighing 1 kg relative to the Oy axis at the time t = 2 seconds. The motion of the point is given as follows: x = 2t, y = t3, z = t4.

To solve the problem, it is necessary to calculate the speed of the point at time t = 2 seconds, using the derivatives of its coordinates with respect to time. Then it is necessary to determine the vector of momentum, which is equal to the product of the mass of the point and its speed. To calculate the angular momentum relative to the Oy axis, it is necessary to project the momentum vector onto this axis and multiply the result by the distance from the Oy axis to the point.

As a result, solving the problem consists of performing the following steps:

1. Calculation of the speed of a point at time t = 2 seconds, using derivatives of its coordinates with respect to time: v = (2, 12, 32).
2. Calculation of the momentum vector: p = mv, where m = 1 kg is the mass of the point.
3. Determination of the angular momentum relative to the Oy axis: L = p × r, where r = (0, 0, 2) is the radius vector of the point relative to the Oy axis. The result is the component of the vector L corresponding to the Oy axis, i.e. Lу.

After completing these steps, we get the answer: Lу = -96.

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