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A belt having dimension 20 cm x 1 cm has the ratio of tension 2 and maximum permissible tension is 140 N/cm2. Weight density of leather is 1000 N/m2. The maximum power that can be transmitted is
- a)2.828 kW
- b)1.828 kW
- c)2.228 kW
- d)3.228 kW
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
A belt having dimension 20 cm x 1 cm has the ratio of tension 2 and ma...
T1 = 140 x 20 = 2800 N
= 3.03 M/sec
Most Upvoted Answer
A belt having dimension 20 cm x 1 cm has the ratio of tension 2 and ma...
Given:
- Dimensions of the belt: 20 cm x 1 cm
- Ratio of tension: 2
- Maximum permissible tension: 140 N/cm2
- Weight density of leather: 1000 N/m2
To find:
The maximum power that can be transmitted.
Solution:
Step 1: Calculation of Maximum Tension:
Given that the maximum permissible tension is 140 N/cm2. We can calculate the maximum tension as follows:
Maximum tension = Maximum permissible tension × Area of the belt
Maximum tension = 140 N/cm2 × (20 cm × 1 cm)
Maximum tension = 140 N/cm2 × 20 cm2
Maximum tension = 2800 N
Step 2: Calculation of Tension:
Given that the ratio of tension is 2. Let the tension in the tight side of the belt be T. Then the tension in the slack side of the belt would be 2T.
Step 3: Calculation of Power:
The power transmitted by a belt is given by the formula:
Power = (T1 - T2) × V
Where,
T1 = Tension in the tight side of the belt
T2 = Tension in the slack side of the belt
V = Velocity of the belt
Step 4: Calculation of Velocity:
The velocity of the belt can be calculated using the formula:
Velocity = (2π × Diameter × N) / 60
Where,
Diameter = Diameter of the pulley
N = Speed of the pulley in rpm
Step 5: Calculation of Diameter:
The diameter of the pulley can be calculated using the formula:
Diameter = (T1 + T2) / (2 × π × Maximum tension)
Step 6: Calculation of Power:
Now, substituting the values of T1, T2, and V in the power formula:
Power = (T1 - T2) × V
Power = (T - 2T) × ((2π × Diameter × N) / 60)
Power = -T × ((2π × Diameter × N) / 60)
Power = -T × ((2π × ((T + 2T) / (2 × π × Maximum tension)) × N) / 60)
Power = -T × ((T + 2T) × N) / (30 × Maximum tension)
Power = -T × (3T × N) / (30 × Maximum tension)
Power = -(T3 × N) / (10 × Maximum tension)
Step 7: Calculation of Maximum Power:
Since Power is maximum when T is maximum, we can differentiate the power equation with respect to T and equate it to zero to find the maximum value.
d(Power) / d(T) = 0
-3T2 × N / (10 × Maximum tension) = 0
T2 = 0
Since the tension cannot be zero, we take the second derivative to confirm that it is a maximum.
d2(Power) / d(T)2 = -6N / (10 × Maximum tension)
Since the second derivative is negative, it confirms that the power is maximum.
Step 8: Final Result:
- Dimensions of the belt: 20 cm x 1 cm
- Ratio of tension: 2
- Maximum permissible tension: 140 N/cm2
- Weight density of leather: 1000 N/m2
To find:
The maximum power that can be transmitted.
Solution:
Step 1: Calculation of Maximum Tension:
Given that the maximum permissible tension is 140 N/cm2. We can calculate the maximum tension as follows:
Maximum tension = Maximum permissible tension × Area of the belt
Maximum tension = 140 N/cm2 × (20 cm × 1 cm)
Maximum tension = 140 N/cm2 × 20 cm2
Maximum tension = 2800 N
Step 2: Calculation of Tension:
Given that the ratio of tension is 2. Let the tension in the tight side of the belt be T. Then the tension in the slack side of the belt would be 2T.
Step 3: Calculation of Power:
The power transmitted by a belt is given by the formula:
Power = (T1 - T2) × V
Where,
T1 = Tension in the tight side of the belt
T2 = Tension in the slack side of the belt
V = Velocity of the belt
Step 4: Calculation of Velocity:
The velocity of the belt can be calculated using the formula:
Velocity = (2π × Diameter × N) / 60
Where,
Diameter = Diameter of the pulley
N = Speed of the pulley in rpm
Step 5: Calculation of Diameter:
The diameter of the pulley can be calculated using the formula:
Diameter = (T1 + T2) / (2 × π × Maximum tension)
Step 6: Calculation of Power:
Now, substituting the values of T1, T2, and V in the power formula:
Power = (T1 - T2) × V
Power = (T - 2T) × ((2π × Diameter × N) / 60)
Power = -T × ((2π × Diameter × N) / 60)
Power = -T × ((2π × ((T + 2T) / (2 × π × Maximum tension)) × N) / 60)
Power = -T × ((T + 2T) × N) / (30 × Maximum tension)
Power = -T × (3T × N) / (30 × Maximum tension)
Power = -(T3 × N) / (10 × Maximum tension)
Step 7: Calculation of Maximum Power:
Since Power is maximum when T is maximum, we can differentiate the power equation with respect to T and equate it to zero to find the maximum value.
d(Power) / d(T) = 0
-3T2 × N / (10 × Maximum tension) = 0
T2 = 0
Since the tension cannot be zero, we take the second derivative to confirm that it is a maximum.
d2(Power) / d(T)2 = -6N / (10 × Maximum tension)
Since the second derivative is negative, it confirms that the power is maximum.
Step 8: Final Result:
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Question Description
A belt having dimension 20 cm x 1 cm has the ratio of tension 2 and maximum permissible tension is 140 N/cm2. Weight density of leather is 1000 N/m2. The maximum power that can be transmitted isa)2.828 kWb)1.828 kWc)2.228 kWd)3.228 kWCorrect answer is option 'A'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A belt having dimension 20 cm x 1 cm has the ratio of tension 2 and maximum permissible tension is 140 N/cm2. Weight density of leather is 1000 N/m2. The maximum power that can be transmitted isa)2.828 kWb)1.828 kWc)2.228 kWd)3.228 kWCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A belt having dimension 20 cm x 1 cm has the ratio of tension 2 and maximum permissible tension is 140 N/cm2. Weight density of leather is 1000 N/m2. The maximum power that can be transmitted isa)2.828 kWb)1.828 kWc)2.228 kWd)3.228 kWCorrect answer is option 'A'. Can you explain this answer?.
A belt having dimension 20 cm x 1 cm has the ratio of tension 2 and maximum permissible tension is 140 N/cm2. Weight density of leather is 1000 N/m2. The maximum power that can be transmitted isa)2.828 kWb)1.828 kWc)2.228 kWd)3.228 kWCorrect answer is option 'A'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A belt having dimension 20 cm x 1 cm has the ratio of tension 2 and maximum permissible tension is 140 N/cm2. Weight density of leather is 1000 N/m2. The maximum power that can be transmitted isa)2.828 kWb)1.828 kWc)2.228 kWd)3.228 kWCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A belt having dimension 20 cm x 1 cm has the ratio of tension 2 and maximum permissible tension is 140 N/cm2. Weight density of leather is 1000 N/m2. The maximum power that can be transmitted isa)2.828 kWb)1.828 kWc)2.228 kWd)3.228 kWCorrect answer is option 'A'. Can you explain this answer?.
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