Question

When two pulleys of different diameters are connected by means of an open belt drive, then the angle of contact taken into consideration should be of the

• A. Large pulley
• B. Smaller pulley
• C. Average of two pulleys
• D. Difference of two pulleys

Hint:

In any system, performance is dictated by the weakest link. In a belt drive, the "weakest link" for friction and power transmission is the pulley with the least amount of belt contact. This is always the smaller pulley in an open belt drive.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question concerns the design and analysis of belt drives. The amount of power a belt can transmit depends on several factors, including the tension in the belt, the coefficient of friction, and the angle of contact (or angle of wrap) between the belt and the pulley. The system's performance is limited by the weakest link.

Step 2: Detailed Explanation:
In an open belt drive connecting two pulleys of different sizes, the belt wraps around a portion of the circumference of each pulley.

• The angle of contact (\(\theta\)) is the angle subtended by the belt at the center of the pulley.

• For the larger pulley, the angle of contact is greater than 180 degrees (\(\pi\) radians).

• For the smaller pulley, the angle of contact is less than 180 degrees (\(\pi\) radians).
The formula for the ratio of tensions in a belt drive is \( \frac{T_1}{T_2} = e^{\mu \theta} \), where \(T_1\) is the tension on the tight side, \(T_2\) is the tension on the slack side, \(\mu\) is the coefficient of friction, and \(\theta\) is the angle of contact.
The maximum power that can be transmitted is proportional to \((T_1 - T_2)\). This difference depends on the ratio \(T_1/T_2\), which in turn depends on \(\theta\).
For the belt to transmit power without slipping, the required frictional grip must be present on both pulleys. The pulley with the smaller angle of contact (\(\theta\)) is more prone to slipping because the term \(e^{\mu \theta}\) will be smaller, meaning a smaller tension ratio can be supported before slip occurs.
Therefore, all design calculations for power transmission capacity must be based on the conditions at the smaller pulley, as it is the critical point where slipping is most likely to begin.

Step 3: Final Answer:
In a belt drive with pulleys of different diameters, the angle of contact on the smaller pulley should be used for design and analysis because it is the limiting factor for power transmission.