### Problem 8-98**Task:**Show that the frictional relationship between the belt tensions, the coefficient of friction \(\mu\), and the angular contacts \(\alpha\) and \(\beta\) for the V-belt is \[ T_2 = T_1 e^{\mu \beta \sin(\alpha/2)} \]**Diagram Explanation:**- **Left Diagram:** - Displays the side view of a V-belt wrapped around a pulley. - Indicates impending motion with an arrow, showing the direction of movement. - Labels \( T_2 \) and \( T_1 \) represent the tensions in the belt on either side of the pulley. - The angle \(\beta\) is shown as the angle subtended by the belt on the pulley, centered at the pivot.- **Right Diagram:** - Provides a cross-sectional view of the V-belt. - Shows the angle \(\alpha\) as the angle of contact between the belt and the pulley sides. This relationship is fundamental in understanding the mechanics of belt drives and their efficiency in transmitting power.

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Answered: 8-98. Show that the frictional…

8-98. Show that the frictional relationship between the belt tensions, the coefficient of friction μ, and the angular contacts a and ß for the V-belt is T₂ = T₁ e sin(a/2)

Elements Of Electromagnetics

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ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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Concept explainers

Basic Terminology in Mechanics

The area of science and mathematics concerned the motion of the objects, the relation between the force matter, and motion if specified. Forces exerted over the objects result in displacements or changes in the position of an object concerning its environment. This branch of physics keeps its roots in Ancient Greece in the form of the scripts of Aristotle and Archimedes.

Question

### Problem 8-98

**Task:**
Show that the frictional relationship between the belt tensions, the coefficient of friction \(\mu\), and the angular contacts \(\alpha\) and \(\beta\) for the V-belt is

\[ T_2 = T_1 e^{\mu \beta \sin(\alpha/2)} \]

**Diagram Explanation:**

- **Left Diagram:**
- Displays the side view of a V-belt wrapped around a pulley.
- Indicates impending motion with an arrow, showing the direction of movement.
- Labels \( T_2 \) and \( T_1 \) represent the tensions in the belt on either side of the pulley.
- The angle \(\beta\) is shown as the angle subtended by the belt on the pulley, centered at the pivot.

- **Right Diagram:**
- Provides a cross-sectional view of the V-belt.
- Shows the angle \(\alpha\) as the angle of contact between the belt and the pulley sides.

This relationship is fundamental in understanding the mechanics of belt drives and their efficiency in transmitting power.

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Step 1: Write the given data

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Step 2: Draw the free body diagram and express dT in terms of N & coefficient of friction:

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Step 3: Express dN in terms of Tension & angle theta:

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Step 4: Equate equation 1 and equation 2 to determine the final expression:

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