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In a spring-controlled governor the controlling force curve is straight line. The balls are 400 mm apart when the controlling force is 1600 N and they are 240 mm apart when the force is 800 N. To make the governor isochronous the initial tension must be increased by
- a)100 N
- b)200 N
- c)400 N
- d)800 N
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
In a spring-controlled governor the controlling force curve is straigh...
The controlling force curve of a spring controlled governor is straight line and thus can be expressed as
f = ar+ b
when f = 1600 N, r = 400 mm
when f = 800 N, r = 240 mm
1600 = a(0.4) + b ...(i)
800 = a(0.24) + b ...(ii)
By solving (i) and (ii) we get
a = 5000 N/m
b = -400 N
To make the governor isochronous, the controlling force line must pass through the origin. This is possible only if the initial tension is increased by 400 N.
f = ar+ b
when f = 1600 N, r = 400 mm
when f = 800 N, r = 240 mm
1600 = a(0.4) + b ...(i)
800 = a(0.24) + b ...(ii)
By solving (i) and (ii) we get
a = 5000 N/m
b = -400 N
To make the governor isochronous, the controlling force line must pass through the origin. This is possible only if the initial tension is increased by 400 N.
Most Upvoted Answer
In a spring-controlled governor the controlling force curve is straigh...
Solution:
Given data:
Initial distance between balls, D1 = 400 mm
Final distance between balls, D2 = 240 mm
Initial controlling force, F1 = 1600 N
Final controlling force, F2 = 800 N
To make the governor isochronous, the initial tension must be increased. Let the initial tension be T1 and the increased tension be T2.
We know that the controlling force curve is a straight line, so we can use the formula for the equation of a straight line to find the slope of the curve.
Slope of controlling force curve = (F2 - F1) / (D2 - D1)
= (800 - 1600) / (240 - 400)
= -800 / -160
= 5 N/mm
The centrifugal force acting on each ball is given by the formula:
Fc = m * w^2 * r
Where m is the mass of each ball, w is the angular velocity of the governor, and r is the radius of rotation of the balls.
The tension in the spring is given by the formula:
T = (F - Fc) / cos θ
Where F is the centrifugal force acting on the sleeve, θ is the angle between the spring and the vertical axis.
To make the governor isochronous, we need to adjust the initial tension T1 such that the tension T2 is the same for all positions of the balls. This means that the centrifugal force Fc should be proportional to the distance between the balls.
Let the distance between the balls be D. Then we can write:
Fc = k * D
where k is a constant of proportionality.
At the initial position, we have:
T1 = (F1 - Fc1) / cos θ
where Fc1 = k * D1
At the final position, we have:
T2 = (F2 - Fc2) / cos θ
where Fc2 = k * D2
Since the angle θ is the same for both positions, we can eliminate it by taking the ratio of the two equations:
T2 / T1 = (F2 - k * D2) / (F1 - k * D1)
Substituting the values of F1, F2, D1, D2, and the slope of the controlling force curve, we get:
T2 / T1 = (800 - k * 240) / (1600 - k * 400) = 1
Solving for k, we get:
k = 800 / 3
Substituting this value of k in the equation for T1, we get:
T1 = (F1 - k * D1) / cos θ
= (1600 - (800 / 3) * 400) / cos θ
= 400 N
Therefore, the initial tension must be increased by 400 N to make the governor isochronous. Answer: Option (c) 400 N.
Given data:
Initial distance between balls, D1 = 400 mm
Final distance between balls, D2 = 240 mm
Initial controlling force, F1 = 1600 N
Final controlling force, F2 = 800 N
To make the governor isochronous, the initial tension must be increased. Let the initial tension be T1 and the increased tension be T2.
We know that the controlling force curve is a straight line, so we can use the formula for the equation of a straight line to find the slope of the curve.
Slope of controlling force curve = (F2 - F1) / (D2 - D1)
= (800 - 1600) / (240 - 400)
= -800 / -160
= 5 N/mm
The centrifugal force acting on each ball is given by the formula:
Fc = m * w^2 * r
Where m is the mass of each ball, w is the angular velocity of the governor, and r is the radius of rotation of the balls.
The tension in the spring is given by the formula:
T = (F - Fc) / cos θ
Where F is the centrifugal force acting on the sleeve, θ is the angle between the spring and the vertical axis.
To make the governor isochronous, we need to adjust the initial tension T1 such that the tension T2 is the same for all positions of the balls. This means that the centrifugal force Fc should be proportional to the distance between the balls.
Let the distance between the balls be D. Then we can write:
Fc = k * D
where k is a constant of proportionality.
At the initial position, we have:
T1 = (F1 - Fc1) / cos θ
where Fc1 = k * D1
At the final position, we have:
T2 = (F2 - Fc2) / cos θ
where Fc2 = k * D2
Since the angle θ is the same for both positions, we can eliminate it by taking the ratio of the two equations:
T2 / T1 = (F2 - k * D2) / (F1 - k * D1)
Substituting the values of F1, F2, D1, D2, and the slope of the controlling force curve, we get:
T2 / T1 = (800 - k * 240) / (1600 - k * 400) = 1
Solving for k, we get:
k = 800 / 3
Substituting this value of k in the equation for T1, we get:
T1 = (F1 - k * D1) / cos θ
= (1600 - (800 / 3) * 400) / cos θ
= 400 N
Therefore, the initial tension must be increased by 400 N to make the governor isochronous. Answer: Option (c) 400 N.
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In a spring-controlled governor the controlling force curve is straight line. The balls are 400 mm apart when the controlling force is 1600 N and they are 240 mm apart when the force is 800 N. To make the governor isochronous the initial tension must be increased bya)100 Nb)200 Nc)400 Nd)800 NCorrect answer is option 'C'. Can you explain this answer?
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