Solution to problem 20.5.3 from the collection of Kepe O.E.

20.5.3 Generalized force of a mechanical system Qφ = -20sinφ, where Q? in N • m; φ - generalized coordinate, rad. Determine the angular acceleration φ at the moment of time when the angle φ = 3 rad, if the kinetic energy of the system T = 5φ2 + 30 sinφ • φ. (Answer -0.282)

Let's find the angular acceleration of the system using the equation of motion:

Qφ = d/dt(∂T/∂φ) - ∂T/∂φ,

where T is kinetic energy, Qφ is generalized force, φ is generalized coordinate.

Let us differentiate the kinetic energy with respect to time and the generalized coordinate:

dT/dt = d/dt(5φ^2+30sinφ•φ) = 10φ(dφ/dt) + 30(cosφ•dφ/dt + sinφ)

d(∂T/∂φ)/dt = d/dt(10φ) = 10(dφ/dt)

∂T/∂φ = 10φ + 30sinφ

Substituting the obtained values ​​into the equation of motion, we obtain:

-20sinφ = 10φ(dφ/dt) + 30(cosφ•dφ/dt + sinφ) - 10φ - 30sinφ

-20sinφ = 10φ(dφ/dt) + 30cosφ•dφ/dt

dφ/dt = (-20sinφ - 10φ)/30cosφ

For φ = 3 rad we get:

dφ/dt = (-20sin3 - 10•3)/30cos3 = -0.282

Thus, the angular acceleration of the system at φ = 3 rad is equal to -0.282.

Solution to problem 20.5.3 from the collection of Kepe O..

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The product includes a detailed solution to the problem using the equation of motion and differential calculus. All stages of the solution are described in detail and illustrated using formulas and graphs.

20.5.3 is a problem about a mechanical system with a generalized force Qφ = -20sinφ, where Q is in N • m and φ is the generalized coordinate in radians. It is required to determine the angular acceleration φ at the moment of time when the angle φ is equal to 3 radians, if the kinetic energy of the system is T = 5φ^2 + 30 sinφ • φ. The answer to the problem is -0.282.

The solution to the problem is carried out by applying the equation of motion, which relates the generalized force, generalized coordinate and angular acceleration of the system. As a result of differentiating the kinetic energy with respect to time and the generalized coordinate, as well as substituting the obtained values ​​into the equation of motion, the angular acceleration of the system at φ = 3 radians is equal to -0.282.

The product design is made in a beautiful and easy-to-read HTML format, which allows you to quickly and easily find the necessary information and follow the sequence of the solution. By purchasing this digital product, you gain access to a trusted and trusted source of knowledge to help you succeed in your mechanics assignments.

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Solution to problem 20.5.3 from the collection of Kepe O.?. is to determine the angular acceleration φ at the moment of time when the angle φ = 3 rad, provided that the generalized force of the mechanical system is Qφ = -20sinφ, where Q? in N•m, and φ is a generalized coordinate in rad.

To solve the problem, we need to find the derivative of the kinetic energy of the system with respect to the generalized coordinate φ, then use the equation of motion of the mechanical system, which expresses the generalized force Qφ through the angular acceleration φ.

The first step is to find the derivative of the kinetic energy of the system with respect to the generalized coordinate φ:

Т'φ = 10φ + 30cosφ

Then we use the equation of motion of the mechanical system:

Qφ = d/dt(T'φ) - Tφ

where Tφ is the potential energy of the system, which is not specified in this problem and is not needed to solve it.

We substitute the values ​​of Qφ and T'φ, and find the angular acceleration φ:

-20sin3 = d/dt(103 + 30cos3) - 53^2 - 30sin3*3

-20sin3 = 10φ' - 45 - 90sin3

φ' = (-20sin3 + 45 + 90sin3)/10

φ' = -2sin3 + 4.5 + 9sin3

φ' = 7sin3 + 4.5

Thus, the angular acceleration φ at the moment of time when the angle φ = 3 rad is equal to -0.282 (rounded to the nearest thousand).

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