Solution to problem 11.2.19 from the collection of Kepe O.E.

11.2.19 The disk rotates around the Oz axis according to the law? = 4 sin 3t Point M moves along its rim according to the equation AM = 0.66 sin 6t + 4. Determine the absolute speed of point M at time t = 0.35 s, if radius R = 1 m. (Answer 3, 97)

To solve this problem, it is necessary to determine the speed of point M at time t = 0.35 s. First of all, it is necessary to determine the angular velocity of the disk, which is determined by the law? = d?/dt, where ? - angle of rotation of the disk in radians, t - time.

From the given rotation law? = 4 sin 3t you can obtain the angular velocity of the disk at time t = 0.35 s by substituting the time value into this expression and making the necessary calculations:

? = 4 sin 3 · 0.35 = 3.28 rad/c.

Next, you need to determine the speed of point M moving in a circle with radius R = 1 meter. To do this, you can use the formula for the speed of a point on a circle: v = R · ?, where v is the speed of the point M, R is the radius of the circle, ? - angular velocity.

Substituting known values, we get:

v = 1 · 3,28 = 3,28 м/c.

However, this speed is relative, since point M moves with the disk. To determine the absolute speed, it is necessary to take into account the movement of point M relative to the disk, which is determined by the given equation of motion AM = 0.66 sin 6t + 4.

At time t = 0.35 s, the AM value will be equal to:

AM = 0.66 sin 6 · 0.35 + 4 = 4.31 m.

Now you can determine the absolute speed of point M using the formula for speed relative to the center of mass:

v_abs = v + w × r,

where w is the angular velocity of the disk, r is the vector directed from the center of the disk to point M.

Vector r has a length R = 1 meter and is directed at an angle ?/2 to the Ox axis, since point M is located at a distance R from the center of the disk and moves in a circle.

Substituting known values, we get:

v_abs = 3,28 + 3,28 × 1 × cos(?/2) = 3,97 м/c.

Thus, the absolute speed of point M at time t = 0.35 s is equal to 3.97 m/s.

Solution to problem 11.2.19 from the collection of Kepe O.?.

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The proposed digital product is a solution to problem 11.2.19 from the collection of Kepe O.?. in electronic format. This file contains a step-by-step explanation of all the calculations and formulas used to solve the problem, as well as beautiful graphics to help you visualize the solution process.

The task is to determine the absolute speed of point M at time t = 0.35 s, if the disk rotates around the Oz axis according to the law? = 4 sin 3t, and point M moves along its rim, corresponding to the equation AM = 0.66 sin 6t + 4. The radius of the disk is R = 1 m.

To solve the problem, it is necessary to calculate the angular velocity of the disk according to the law of rotation, then the speed of point M moving in a circle with radius R, and, finally, the absolute speed of point M, taking into account its movement relative to the disk.

The digital product is available for download immediately after purchase in a convenient PDF format that can be opened on any device that supports reading PDF files. This digital product will be useful for both students and teachers who are looking for good challenges for their students.

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The offered product is a solution to problem 11.2.19 from the collection "Problems for the general course of physics" by Kepe O.?.

The problem considers a disk rotating around the vertical axis Oz according to a given law of motion. On the rim of the disk there is a point M, moving according to a given equation. It is necessary to find the absolute speed of point M at a given time t=0.35 s, if the radius of the disk R=1 meter.

The solution to the problem is to find the speed of point M at time t=0.35 s. To do this, you need to calculate the derivative of the AM equation with respect to time, then substitute the values ​​of t and R and get the answer.

The answer to the problem is 3.97 m/s.

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