Solution to problem 8.4.4 from the collection of Kepe O.E.

8.4.4. Calculating the speed of a point on a gear

Let's consider gear 1 of radius R1 = 0.4 m, which rotates uniformly with angular acceleration ?1 = 4 rad/s². Wheel 1 is connected to gear 3 of radius R3 = 0.5 m. The movement begins from a state of rest.

It is necessary to determine the speed of point M at time t = 2 s.

Answer:

Let's calculate the angular acceleration of wheel 1 after time t = 2 s:

?1 = ?01*t + (1/2)at²

where a is the angular acceleration, ?01 is the initial angular velocity.

Since the initial state is at rest, then ?01 = 0.

Then we get:

?1 = (1/2)αt² = 4 rad/s²

From here we find the angular acceleration:

α = (2*?1)/t² = 8 rad/s³

Since wheel 1 is connected to wheel 3, the angular velocities of these wheels are equal:

?1R1 = ?3R3

From here we obtain the angular velocity of wheel 3:

?3 = (?1*R1)/R3 = 3.2 rad/s

The speed of point M on wheel 3 can be found using the formula:

V = ?3*R3

Then the speed of point M at time t = 2 s is equal to:

V = 3.2 * 0.5 = 1.6 m/s

Thus, the speed of point M on the gear wheel at time t = 2 s is 1.6 m/s.

Solution to problem 8.4.4 from the collection of Kepe O.?.

We present to your attention the solution to problem 8.4.4 from the collection of Kepe O.?. This digital product is a solution to a physics problem involving calculating the speed of a point on a gear. The solution is completed by a professional teacher and is accompanied by a detailed description of each step of the calculation. You can purchase this product in our digital goods store and use it to prepare for exams on your own, as well as to practice your physics problem-solving skills.

Price: 150 rubles.

Offered for sale is a solution to problem 8.4.4 from the collection of Kepe O.?. on physics related to calculating the speed of a point on a gear wheel. The problem considers two gears: wheel 1 of radius R1 = 0.4 m and wheel 3 of radius R3 = 0.5 m, connected to each other. Wheel 1 rotates uniformly with angular acceleration ?1 = 4 rad/s², while the movement begins from a state of rest. The task is to determine the speed of point M on wheel 3 at time t = 2 s.

Solving the problem consists of several steps. First, the angular acceleration of wheel 1 is found after time t = 2 s, then the angular velocity of wheel 3 is found using the relationship between the angular velocities of the gears. Finally, the speed of point M on wheel 3 is found.

The solution is completed by a professional teacher and is accompanied by a detailed description of each step of the calculation. The solution can be used to independently prepare for exams and practice problem-solving skills in physics. The price of the product is 150 rubles.

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Solution to problem 8.4.4 from the collection of Kepe O.?. is associated with calculating the speed of point M on a gear wheel that rotates uniformly. To solve the problem, it is necessary to know the angular acceleration of the gear and the radii of the gears.

In this problem, the angular acceleration of the gear is ?1 = 4 rad/s2, and the radii of the gears are R1 = 0.4 m and R3 = 0.5 m. The movement begins from a state of rest.

To determine the speed of point M at time t = 2 s, it is necessary to apply the relationship between linear speed and angular speed:

v = ? * R,

where v is linear speed, ? - angular velocity, R - radius of the gear.

To determine the angular velocity of the gear at time t = 2 s, it is necessary to use the formula:

? = ?0 + ?1 * t,

where ?0 is the initial angular velocity, ?1 is the angular acceleration, t is time.

From the conditions of the problem it follows that the initial angular velocity is zero, since the movement begins from a state of rest.

Thus, to determine the speed of point M at time t = 2 s, it is necessary to calculate the angular velocity of the gear at time t = 2 s and substitute it into the formula for linear speed.

Let's calculate the angular velocity of the gear at time t = 2 s:

? = ?0 + ?1 * t = 0 + 4 * 2 = 8 rad/s.

Now we can calculate the linear speed of point M:

v = ? * R3 = 8 * 0.5 = 4 m/s.

Thus, the speed of point M at time t = 2 s is 4 m/s. Answer: 3.2.

Problem 8.4.4 from the collection of Kepe O.?. solves the following problem: there is a cube with side a inscribed in a sphere. Find the volume between the cube and the sphere.

The solution to this problem begins with calculating the radius of the sphere into which the cube is inscribed. This can be done knowing that the diagonal of the cube is equal to the diameter of the inscribed sphere. Thus, the radius of the sphere will be equal to a/√2.

Then you need to calculate the volume of the cube and the volume of the sphere. The volume of a cube is equal to a^3, and the volume of a sphere can be calculated using the formula V = (4/3)πr^3, where r is the radius of the sphere. We get:

V_cube = a^3 V_spheres = (4/3)π(a/√2)^3

Finally, the volume of the desired figure will be equal to the difference between the volume of the sphere and the volume of the cube:

V_shapes = V_spheres - V_cube = (4/3)π(a/√2)^3 - a^3/1

All that remains is to substitute the numerical values ​​and calculate the answer.

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