The **question aims to find the tension** in a rope having some weight in different conditions when the** box is at rest,** **moving with constant velocity,** and moving with some value of **speed and acceleration**. **Tension **is defined as the force transmitted by a rope, string, or wire when** pulled by forces acting from opposite sides**. The **pulling force** is directed along the length of the wire, pulling energy evenly onto the **bodies at the ends**.

**For example**, if a person pulls on an **immaterial rope** with a force of $40\: N$, a force of $40\: N$ also acts on the block. All immaterial ropes are subject to two opposite and equal tension forces. Here, a **person is pulling a block with a rope**, so the rope experiences a net force. Therefore, two opposing and equal tensile forces act on all massless strings. **When a person pulls on a block, the rope experiences tension in one direction from the pull and tension in the other direction from the reactive force of the block.**

The** formula for the tension** in the rope is:

\[T=ma+mg\]

Where $T$ is the **tension**, $m$ is the **mass**, $a$ is the **acceleration**, and $g$ is the **gravitational force.**

**Expert Answer**

**Given data**: $50\:kg$

**Part(a)**

The **box is at rest,** that is, it is not moving, the **acceleration is zero** if it is accelerated by zero, the **sum of all the forces acting on the box is zero.**

According to Newton’s second law of motion:

\[F=ma\]

\[F=m.(0 \dfrac{m}{s^{2}})\]

\[F=0\:N\]

\[T_{1}=0\:N\]

**Part(b)**

\[v=5\dfrac{m}{s}\]

The **box moves at a constant speed**. The **acceleration is zero** in this case.

\[F=ma\]

\[F=m.(0 \dfrac{m}{s^{2}})\]

\[F=0\:N\]

\[T_{2}=0\:N\]

**Part(c)**

\[v_{x}=5\dfrac{m}{s}\]

\[a_{x}=5\dfrac{m}{s^{2}}\]

**Acceleration is not zero** in this case.

\[F=ma\]

\[F=(50\:kg)(5\dfrac{m}{s^{2}})\]

\[F=250\:N\]

\[T_{3}=250\:N\]

**Numerical Result**

The **tension in the rope** when the **box is at rest** is:

\[T_{1}=0\:N\]

The **tension in the rope** when the box moves at a **steady speed** is:

\[T_{2}=0\:N\]

The **tension in the rope when the box moves with velocity** $v_{x}=5\dfrac{m}{s}$ and **acceleration** $a_{x}=5\dfrac{m}{s^{2}}$ is:

\[T_{3}=250\:N\]

**Example**

**A horizontal rope is tied to a $60\:kg$ crate on frictionless ice. What is the tension in the rope if:**

**Part(a) Is the box at rest?**

**Part(b) Is the box moving at a constant speed of $10.0\: m/s$?**

**Part(c) The box has $v_{x}=10\dfrac{m}{s}$ and acceleration $a_{x}=10\dfrac{m}{s^{2}}$**

**Solution**

**Given data**: $60\:kg$

**Part(a)**

The **box is at rest,** that is, it is not moving, the **acceleration is zero** if it is accelerated by zero, the **sum of all the forces acting on the box is zero.**

According to Newton’s second law of motion:

\[F=ma\]

\[F=m.(0 \dfrac{m}{s^{2}})\]

\[F=0\:N\]

\[T_{1}=0\:N\]

**Part(b)**

\[v=10\dfrac{m}{s}\]

The **box moves at a constant speed**. The **acceleration is zero** in this case.

\[F=ma\]

\[F=m.(0 \dfrac{m}{s^{2}})\]

\[F=0\:N\]

\[T_{2}=0\:N\]

**Part(c)**

\[v_{x}=10\dfrac{m}{s}\]

\[a_{x}=10\dfrac{m}{s^{2}}\]

**Acceleration is not zero** in this case.

\[F=ma\]

\[F=(60\:kg)(10\dfrac{m}{s^{2}})\]

\[F=600\:N\]

\[T_{3}=600\:N\]

The **tension in the rope** when the **box is at rest** is:

\[T_{1}=0\:N\]

The **tension in the rope** when the box moves at a **steady speed** is:

\[T_{2}=0\:N\]

The **tension in the rope when the box moves with velocity** $v_{x}=10\dfrac{m}{s}$ and **acceleration** $a_{x}=10\dfrac{m}{s^{2}}$ is:

\[T_{3}=600\:N\]