The manipulator uniformly accelerates the mass in 2 s

In 2 s, the manipulator uniformly accelerates a load weighing 5 kg in an arc whose radius is 1.5 m. Determine the maximum drive power of the manipulator if it is known: the moment of inertia of the manipulator is 15 kg * m2; rotation angle 90°; the load can be considered a point mass.

Problem 10921. Detailed solution with a brief record of the conditions, formulas and laws used in the solution, derivation of the calculation formula and answer. If you have any questions regarding the solution, please write. I try to help.

Product description:

A manipulator is a device designed to move loads. This manipulator is capable of uniformly accelerating a load weighing 5 kg in an arc with a radius of 1.5 meters in 2 seconds. For this manipulator, the moment of inertia is known, which is equal to 15 kg * m2, and the rotation angle, which is 90 degrees. The load can be considered a point mass.

To determine the maximum drive power of the manipulator, it is necessary to use the following formulas and laws:

1. Law of uniformly varied motion: s = (v0 + v)t/2, where s is the path, v0 is the initial speed, v is the final speed, t is the time of movement.

2. Law of conservation of energy: E1 + W = E2, where E1 is the initial energy, W is the work, E2 is the final energy.

3. Power: P = W/t, where P is power, W is work, t is time.

To calculate the maximum power of the manipulator drive, it is necessary to determine the work performed by the drive when moving the load along an arc. The work is calculated as the difference between the initial and final energy of the system, that is:

W = E2 - E1

To calculate the initial and final energy of the system, the following formulas must be used:

E1 = 0 (since the initial speed and height are zero)

E2 = mv2/2 + Iω2/2, where m is the mass of the load, v is the speed of the load, I is the moment of inertia of the manipulator, ω is the angular velocity.

To calculate angular velocity, use the following formula:

ω = αt, where α is angular acceleration, t is time.

Angular acceleration can be expressed in terms of linear acceleration using the following formula:

α = a/r, where a is linear acceleration, r is the radius of the arc.

Then the angular velocity will be equal to:

ω = a*t/r

Knowing the speed of the load v and the angular speed ω, we can find the drive power of the manipulator:

P = W/t = (mv2/2 + Iω2/2)/t

Thus, the maximum drive power of the manipulator will be equal to the power value at maximum load speed. To find this value, you need to find the maximum value of the linear acceleration of the load. For uniformly accelerated motion, this value is equal to the acceleration, which can be found from the following formula:

a = v/t

Then the maximum speed of the load will be equal to:

v = a*t

Substituting the found values ​​into the formula for the drive power of the manipulator, we get the answer:

P = (mv2/2 + Ia2t2/2)/t = (5*(at)2/2 + 15(at/r)2/2)/2 = 37.5a2/r2, where a = v/t and v = a*t.

Thus, the maximum drive power of the manipulator is 37.5*(v/t)2/1.52 W.

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