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# Tension **Definition**: Tension is the force exerted by a rope, string, cable, or similar object on another object. It is a pulling force that acts along the length of the rope and is directed away from the object it is attached to. ## Key Points * Tension is a pulling force. * Tension acts...

# Tension **Definition**: Tension is the force exerted by a rope, string, cable, or similar object on another object. It is a pulling force that acts along the length of the rope and is directed away from the object it is attached to. ## Key Points * Tension is a pulling force. * Tension acts along the length of the rope. * Tension is directed away from the object. ## Tension in a Rope Consider a rope attached to a block being pulled horizontally. The tension in the rope is the force exerted by the rope on the block. * If the rope is massless and inextensible (not stretching), the tension is the same throughout the rope. * If the rope has mass, the tension may vary along the rope. * If the rope is accelerating, the tension may not be the same throughout the rope. ## Examples ### Example 1 A block of mass $m$ is attached to a rope and pulled horizontally with a force $F$. What is the tension in the rope? **Solution**: The tension in the rope is equal to the applied force $F$. $T = F$ ### Example 2 A block of mass $m$ is hanging vertically from a rope. What is the tension in the rope? **Solution**: The tension in the rope is equal to the weight of the block. $T = mg$ where $g$ is the acceleration due to gravity. ### Example 3 Two blocks of masses $m_1$ and $m_2$ are connected by a rope passing over a pulley. Find the tension in the rope. **Solution**: Let $m_2 > m_1$. The equations of motion for the two blocks are: $T - m_1g = m_1a$ $m_2g - T = m_2a$ Adding the two equations gives: $m_2g - m_1g = m_1a + m_2a$ $a = \frac{m_2 - m_1}{m_1 + m_2}g$ Substituting the value of $a$ in the first equation: $T = m_1g + m_1a = m_1g + m_1\frac{m_2 - m_1}{m_1 + m_2}g$ $T = m_1g(1 + \frac{m_2 - m_1}{m_1 + m_2}) = m_1g(\frac{m_1 + m_2 + m_2 - m_1}{m_1 + m_2})$ $T = \frac{2m_1m_2}{m_1 + m_2}g$ ### Example 4 A block of mass $m$ is placed on an inclined plane with an angle $\theta$. The block is attached to a rope. Find the tension in the rope required to keep the block in equilibrium (not moving). **Solution**: The forces acting on the block are: * Weight $mg$ acting vertically downwards. * Normal reaction $N$ acting perpendicular to the inclined plane. * Tension $T$ acting along the inclined plane. Resolving the weight into components parallel and perpendicular to the inclined plane: Component parallel to the inclined plane = $mg\sin\theta$ Component perpendicular to the inclined plane = $mg\cos\theta$ For equilibrium: $T = mg\sin\theta$ $N = mg\cos\theta$ ### Example 5 A car is pulling a trailer with a rope. The mass of the car is $m_1$ and the mass of the trailer is $m_2$. The force exerted by the car is $F$. What is the tension in the rope connecting the car and the trailer? **Solution**: The acceleration of the car and the trailer is: $a = \frac{F}{m_1 + m_2}$ The tension in the rope is the force required to accelerate the trailer: $T = m_2a = m_2\frac{F}{m_1 + m_2}$