To calculate the tension in the string, we can apply Newton's second law for both the object hanging from the pulley and the pulley itself.

Given:

• Mass of the pulley, $m_p = 1.50 \, \text{kg}$
• Radius of the pulley, $r = 0.270 \, \text{m}$
• Mass of the hanging object, $m = 0.270 \, \text{kg}$
• Acceleration due to gravity, $g = 9.81 \, \text{m/s}^2$

Step 1: Analyze the forces on the hanging mass

For the hanging mass $m$: $F_{\text{net}} = mg - T = ma$ where $T$ is the tension in the string and $a$ is the linear acceleration of the mass.

Step 2: Analyze the pulley

The torque $\tau$ on the pulley due to the tension in the string is given by: $\tau = T \cdot r$ The torque is also related to the angular acceleration $\alpha$ of the pulley: $\tau = I \alpha$ where $I$ is the moment of inertia of the pulley. For a solid cylinder: $I = \frac{1}{2} m_p r^2$ The relationship between linear acceleration $a$ and angular acceleration $\alpha$ is: $a = r \alpha \quad \Rightarrow \quad \alpha = \frac{a}{r}$

Step 3: Substitute into the torque equation

Substituting for $\alpha$: $T \cdot r = I \cdot \frac{a}{r}$ Substituting $I$: $T \cdot r = \left(\frac{1}{2} m_p r^2\right) \cdot \frac{a}{r}$ This simplifies to: $T = \frac{1}{2} m_p a$

Step 4: Substitute into the force equation for the hanging mass

Now substituting $T$ into the first equation: $mg - \frac{1}{2} m_p a = ma$ Rearranging gives: $mg = ma + \frac{1}{2} m_p a$ Factoring out $a$: $mg = a \left( m + \frac{1}{2} m_p \right)$ Solving for $a$: $a = \frac{mg}{m + \frac{1}{2} m_p}$

Step 5: Calculate $a$

Substituting the known values: $a = \frac{0.270 \cdot 9.81}{0.270 + \frac{1}{2} \cdot 1.50}$ Calculating: $a = \frac{2.6457}{0.270 + 0.75} = \frac{2.6457}{1.020} \approx 2.594 \, \text{m/s}^2$

Step 6: Calculate the tension $T$

Now substituting back to find $T$: $T = \frac{1}{2} m_p a = \frac{1}{2} \cdot 1.50 \cdot 2.594 \approx 1.938 \, \text{N}$

Final Answer

The tension in the string is approximately: $\boxed{1.94 \, \text{N}}$

Do you want further details on any step or have any questions? Here are some related questions you might consider:

1. How would the tension change if the mass of the hanging object increased?
2. What would happen to the acceleration if the radius of the pulley was smaller?
3. How does the moment of inertia affect the rotational motion of the pulley?
4. If the pulley were frictionless, how would that affect the tension and acceleration?
5. What assumptions were made in this analysis regarding the string and pulley?

Tip: Always ensure to check unit consistency when performing calculations.