Solution to problem 11.4.12 from the collection of Kepe O.E.

11.4.12. Question about the movement of a point on a disk.

Let us assume that point M moves along a disk of radius R with linear speed v. Then its angular velocity will be equal to w = v / R. When the disk rotates with angular acceleration a, the angular velocity of point M will change with time and equal w = 3 + at.

In order to find the required speed of point M, at which the Coriolis acceleration will be equal to 20 m/s², we use the Coriolis acceleration formula: Fк = 2m(v × w), where m is the mass of point M, v is the linear speed of the point, w is the angular speed disk rotation speed.

Substituting the values ​​and taking into account that Fк = 20 m/s² and w = 3 + at, we get:

20 = 2mv(3 + at)

We also know that the linear velocity v of a point M is equal to the product of its angular velocity w and the radius R of the disk:

v = wR

Substituting this expression into the equation for the Coriolis acceleration, we obtain:

20 = 6mR + 2matR²

Now we can find the required speed v:

v = wR = (3 + at)R

Substituting this expression into the equation for the Coriolis acceleration, we obtain:

20 = 6mR + 2maR²t

Expressing t, we get:

t = (20 - 6mR) / (2maR²)

Now we substitute the found value of t into the expression for speed v:

v = (3 + at)R = (3 + a(20 - 6mR) / (2maR²))R = 3R + (10 - 3mR) / a

Thus, for the Coriolis acceleration of point M to be equal to 20 m/s² at time t = 1 s, it is necessary for point M to move along the rim of the disk with a speed v = 2 m/s (answer 2).

Solution to problem 11.4.12 from the collection of Kepe O.?.

This digital product is the solution to problem 11.4.12 from the collection of Kepe O.?. in theoretical mechanics. The solution is presented in the form of a detailed description with a step-by-step explanation of solution methods and mathematical calculations.

This problem considers the motion of a point on a disk that rotates uniformly accelerated around the Oz axis with angular acceleration and initial angular velocity. You will need to find the speed of point M along the rim of the disk so that at time t = 1 s the Coriolis acceleration of this point is equal to 20 m/s².

The solution to the problem is presented in pdf e-book format, which you can easily download and use to prepare for exams or improve your skills in the field of theoretical mechanics.

Buy the solution to problem 11.4.12 from the collection of Kepe O.?. and get a high-quality product that will help you solve theoretical mechanics problems easily and successfully.

Product description: this is the solution to problem 11.4.12 from the collection of Kepe O.?. in theoretical mechanics. The problem considers the motion of a point on a disk of radius R, which rotates uniformly accelerated around the Oz axis with angular acceleration a and an initial angular velocity of 3 rad/s. It is necessary to find the speed of point M along the rim of the disk so that at time t = 1 s the Coriolis acceleration of this point is equal to 20 m/s². The solution to the problem is presented in e-book format in pdf format with a detailed description of solution methods and mathematical calculations. This product will help you easily and successfully solve problems in theoretical mechanics and prepare for exams or improve your skills in this field.

We present to your attention the solution to problem 11.4.12 from the collection of Kepe O.?. in theoretical mechanics.

According to the conditions of the problem, point M moves along a disk of radius R with linear speed v. The angular velocity of point M will be equal to w = v / R. When the disk rotates with angular acceleration a, the angular velocity of point M will change with time and equal w = 3 + at.

In order to find the required speed of point M, at which the Coriolis acceleration will be equal to 20 m/s², we use the Coriolis acceleration formula: Fк = 2m(v × w), where m is the mass of point M, v is the linear speed of the point, w is the angular speed disk rotation speed.

Substituting the values ​​and taking into account that Fк = 20 m/s² and w = 3 + at, we get:

20 = 2mv(3 + at)

We also know that the linear velocity v of a point M is equal to the product of its angular velocity w and the radius R of the disk:

v = wR

Substituting this expression into the equation for the Coriolis acceleration, we obtain:

20 = 6mR + 2matR²

Now we can find the required speed v:

v = wR = (3 + at)R

Substituting this expression into the equation for the Coriolis acceleration, we obtain:

20 = 6mR + 2maR²t

Expressing t, we get:

t = (20 - 6mR) / (2maR²)

Now we substitute the found value of t into the expression for speed v:

v = (3 + at)R = (3 + a(20 - 6mR) / (2maR²))R = 3R + (10 - 3mR) / a

Thus, for the Coriolis acceleration of point M to be equal to 20 m/s² at time t = 1 s, it is necessary for point M to move along the rim of the disk with a speed v = 2 m/s (answer 2).

The solution to the problem is presented in pdf e-book format, which you can easily download and use to prepare for exams or improve your skills in the field of theoretical mechanics. Buy the solution to problem 11.4.12 from the collection of Kepe O.?. and get a high-quality product that will help you solve theoretical mechanics problems easily and successfully.

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The product in this case is the solution to problem 11.4.12 from the collection of Kepe O.?. The task is formulated as follows:

The disk rotates uniformly accelerated around the Oz axis with angular acceleration α = 2 rad/s^2. It is necessary to determine the speed v of point M on the rim of the disk, at which the Coriolis acceleration of this point will be equal to 20 m/s^2 at time t = 1 s, if the initial angular velocity of the disk is ω_0 = 3 rad/s.

To solve the problem, it is necessary to use the Coriolis equation, which expresses the Coriolis acceleration through the speed and angular velocity of rotation of the observed point in the inertial reference frame. After finding the Coriolis acceleration, you can write an equation to determine the speed v of point M on the rim of the disk.

The solution to this problem contains several stages: finding the angular velocity of rotation of the disk at time t, calculating the Coriolis acceleration of point M at a given speed v, and finding v from the equation connecting the Coriolis acceleration and speed.

The final answer to the problem is 2 m/s.

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