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A flywheel has a mass of 300 kg and a radius of gyration of 1m. It is given a spin of 100 r.p.m about its horizontal axis. The whole assembly rotates about a vertical axis at 6 rad/sec. The gyroscopic couple experienced will be
- a)3π kNm
- b)6π kNm
- c)180π kNm
- d)360π kNm
Correct answer is option 'B'. Can you explain this answer?
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A flywheel has a mass of 300 kg and a radius of gyration of 1m. It is ...
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A flywheel has a mass of 300 kg and a radius of gyration of 1m. It is ...
Given:
Mass of the flywheel, m = 300 kg
Radius of gyration, k = 1 m
Spin speed about horizontal axis, ω₁ = 100 rpm
Rotation speed about vertical axis, ω₂ = 6 rad/sec
To find:
The gyroscopic couple experienced by the flywheel.
Solution:
Step 1: Calculate the spin angular velocity
The spin angular velocity, ω₁, is given in rpm. To convert it to rad/sec, we use the following formula:
ω = (2π/60) * ω₁
Substituting the given value of ω₁, we get:
ω = (2π/60) * 100
ω = (π/30) * 100
ω = (10π/3) rad/sec
Step 2: Calculate the moment of inertia (I)
The moment of inertia of the flywheel can be calculated using the formula:
I = mk²
Substituting the given values of mass (m) and radius of gyration (k), we get:
I = 300 * 1²
I = 300 kg·m²
Step 3: Calculate the gyroscopic couple (C)
The gyroscopic couple (C) can be calculated using the formula:
C = I * ω₁ * ω₂
Substituting the values of moment of inertia (I), spin angular velocity (ω₁), and rotation speed (ω₂), we get:
C = 300 * (10π/3) * 6
C = 6000π N·m
C ≈ 18850 N·m
Step 4: Convert the gyroscopic couple to kNm
To convert the gyroscopic couple from N·m to kNm, we divide by 1000:
C = 18850 N·m
C ≈ 18.85 kNm
Therefore, the gyroscopic couple experienced by the flywheel is approximately 18.85 kNm.
Mass of the flywheel, m = 300 kg
Radius of gyration, k = 1 m
Spin speed about horizontal axis, ω₁ = 100 rpm
Rotation speed about vertical axis, ω₂ = 6 rad/sec
To find:
The gyroscopic couple experienced by the flywheel.
Solution:
Step 1: Calculate the spin angular velocity
The spin angular velocity, ω₁, is given in rpm. To convert it to rad/sec, we use the following formula:
ω = (2π/60) * ω₁
Substituting the given value of ω₁, we get:
ω = (2π/60) * 100
ω = (π/30) * 100
ω = (10π/3) rad/sec
Step 2: Calculate the moment of inertia (I)
The moment of inertia of the flywheel can be calculated using the formula:
I = mk²
Substituting the given values of mass (m) and radius of gyration (k), we get:
I = 300 * 1²
I = 300 kg·m²
Step 3: Calculate the gyroscopic couple (C)
The gyroscopic couple (C) can be calculated using the formula:
C = I * ω₁ * ω₂
Substituting the values of moment of inertia (I), spin angular velocity (ω₁), and rotation speed (ω₂), we get:
C = 300 * (10π/3) * 6
C = 6000π N·m
C ≈ 18850 N·m
Step 4: Convert the gyroscopic couple to kNm
To convert the gyroscopic couple from N·m to kNm, we divide by 1000:
C = 18850 N·m
C ≈ 18.85 kNm
Therefore, the gyroscopic couple experienced by the flywheel is approximately 18.85 kNm.
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A flywheel has a mass of 300 kg and a radius of gyration of 1m. It is given a spin of 100 r.p.m about its horizontal axis. The whole assembly rotates about a vertical axis at 6 rad/sec. The gyroscopic couple experienced will bea)3π kNm b)6π kNm c)180π kNm d)360π kNmCorrect answer is option 'B'. Can you explain this answer?
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