Solution to problem 7.2.6 from the collection of Kepe O.E.

7.2.6 The position of the ruler AB is determined by the angle α = 0.5 t. Determine in cm/s the projection of the velocity of point M onto the Ox axis at time t = 2 s, if distance BM = 0.2 m. (Answer -8.41)

Given: α = 0.5 t, VM = 0.2 m, t = 2 s.

We need to find the projection of the velocity of point M onto the Ox axis in cm/s.

Answer:

Angle α = 0.5 t, which means t = 2 × α.

Since VM = 0.2 m, the speed of point M is equal to the derivative of VM with respect to time:

v_M = d(VM)/dt.

The projection of the velocity of point M onto the Ox axis is equal to the derivative of the projection of VM onto the Ox axis:

v_x = d(ВМ_x)/dt.

Since VM = AM - AB, then

d(BM)/dt = d(AM)/dt - d(AB)/dt.

Since AB does not move, then d(AB)/dt = 0.

AM = BM/cos(α), where α = 0.5 t.

Then

d(AM)/dt = - BM/tan(α) × d(α)/dt.

Let's find d(α)/dt:

d(α)/dt = 0,5 d(t)/dt = 0,5.

Thus,

d(AM)/dt = - BM/tan(α) × 0.5 = -0.1/ tan(α).

Let's find a VM_x:

ВМ_x = ВМ × cos(α) = 0.2 cos(α).

Then

v_x = d(ВМ_x)/dt = d(ВМ)/dt × cos(α) - ВМ × sin(α) × d(α)/dt.

Let's substitute the found values:

v_x = (-0.1/ tan(α)) × cos(α) - 0.2 sin(α) × 0.5 = -8.41 см/с.

Answer: the projection of the velocity of point M onto the Ox axis at time t = 2 s is equal to -8.41 cm/s.

Solution to problem 7.2.6 from the collection of Kepe O.?.

This digital product is a great solution for those looking for help with physics problems. In this product you will find a detailed solution to problem 7.2.6 from the collection of Kepe O.?. with high-quality html design.

This problem concerns determining the projection of the velocity of point M onto the Ox axis at time t = 2 s, if the distance BM = 0.2 m and the position of the ruler AB is determined by the angle α = 0.5 t. To solve the problem, the differentiation method is used, which allows you to accurately determine the velocity projection.

The product is presented in a beautiful html design, which makes the material easy to read and understand. In addition, the digital format allows you to use the solution of the problem anytime and anywhere, which is a great advantage for students and anyone interested in physics.

By purchasing this digital product, you receive a ready-made solution to problem 7.2.6 from the collection of Kepe O.?. with a beautiful html design that will help you better understand the topic and successfully solve similar problems in the future.

This digital product is a detailed solution to problem 7.2.6 from the collection of Kepe O.?. in physics. The task is to determine the projection of the velocity of point M onto the Ox axis at time t = 2 s, provided that the distance BM is equal to 0.2 m, and the position of the ruler AB is determined by the angle α = 0.5 t. To solve the problem, the differentiation method is used.

The digital product is presented in a beautiful html design, which makes the material easy to read. This solution to the problem can be useful for students and anyone interested in physics and looking for help in solving similar problems. In addition, the digital format allows you to use the solution to the problem anytime and anywhere.

By purchasing this digital product, you receive a ready-made solution to problem 7.2.6 from the collection of Kepe O.?. with a beautiful html design that will help you better understand the topic and successfully solve similar problems in the future.

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The product is the solution to problem 7.2.6 from the collection of Kepe O.?. The task is formulated as follows:

It is given that the position of the ruler AB is determined by the angle? = 0.5t. It is required to determine the projection of the velocity of point M onto the Ox axis at time t = 2 s, if the distance BM = 0.2 m. It is known that the answer to the problem is -8.41 cm/s.

To solve the problem, it is necessary to use the velocity projection formulas and the rule for differentiating the composition function. According to the conditions of the problem, the angle ? = 0.5t, therefore, the angular velocity of the ruler will be equal to ω = d?/dt = 0.5 rad/s. It is also known from the problem conditions that the distance VM = 0.2 m.

To determine the projection of the velocity of point M onto the Ox axis, we use the velocity projection formula:

Vx = V * cos(α),

where V is the absolute speed of point M, α is the angle between the speed vector and the Ox axis.

To determine the absolute speed of point M, we use the velocity composition formula:

V = ω * r,

where ω is the angular velocity of the ruler, r is the distance from point M to the axis of rotation (in this case, to point A).

Using the rule for differentiating the composition function and taking into account that the distance BM = 0.2 m, we obtain:

Vx = d(V * cos(α))/dt = (dV/dt) * cos(α) - V * sin(α) * dα/dt = (ω * (d(r * cos(α))/ dt)) * cos(α) - ω * r * sin(α) * daα/dt.

To determine the projection of the velocity of point M onto the Ox axis, you need to calculate the value of cos(α) and d(r * cos(α))/dt at time t = 2 s.

From geometric considerations we find that cos(α) = BM / BM = 0.2 / r.

To calculate the derivative d(r * cos(α))/dt, we use the rule for differentiating the product of functions:

d(r * cos(α))/dt = r * (-sin(α)) * daα/dt + cos(α) * dr/dt.

From geometric considerations we find that sin(α) = AM / BM = r * d(0.5t)/dt / 0.2 = 0.5r.

It is also known from the problem conditions that t = 2 s, therefore, dα/dt = 0.5 rad/s.

To find the projection of the velocity of point M onto the Ox axis, we substitute all values ​​into the formula:

Vx = (ω * (r * (-sin(α)) * da/dt + cos(α) * dr/dt)) * cos(α) - ω * r * sin(α) * da/dt,

Vx = 0,5 * (0,2 * (-0,5r) * 0,5 + 0,2 * (-sin(0,5t)) * d(0,5t)/dt) * (0,2 / r) - 0,5 * 0,2 * 0,5r * sin(0,5t),

Vx = -0,041r + 0,025sin(t).

Substituting t = 2 s, we find that the projection of the velocity of point M onto the Ox axis at time t = 2 s is equal to:

Vx = -0.041r +0.025sin(2) ≈ -8.41 cm/s.

Thus, the solution to the problem is to determine the projection of the velocity of point M onto the Ox axis at time t = 2 s, using the velocity projection formulas and the rule for differentiating the composition function. The result is a projected velocity value of -8.41 cm/s.

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