To calculate the acceleration of the cylindrical object, we can use the formula for rotational motion:
\[a = \frac{g \cdot M}{M + m}\]
where:
- \(a\) is the acceleration of the object
- \(g\) is the acceleration due to gravity (approximately 9.8 m/s²)
- \(M\) is the mass of the cylindrical object (3.97 g or 0.00397 kg)
- \(m\) is the mass of the hanging bucket (5.3 g or 0.0053 kg)
Substituting the given values into the formula, we get:
\[a = \frac{9.8 \cdot 0.00397}{0.00397 + 0.0053} = \frac{0.038806}{0.00927} \approx 4.19 \, \text{m/s²}\]
The acceleration of the cylindrical object is approximately \(4.19 \, \text{m/s²}\).
To calculate the tension in the string, we can use Newton's second law of motion for rotation:
\[T - mg = I \cdot \alpha\]
where:
- \(T\) is the tension in the string
- \(m\) is the mass of the hanging bucket (0.0053 kg)
- \(g\) is the acceleration due to gravity (9.8 m/s²)
- \(I\) is the moment of inertia of the cylindrical object (for a solid cylinder, \(I = \frac{1}{2}MR^2\))
- \(\alpha\) is the angular acceleration (which is related to linear acceleration by \(\alpha = \frac{a}{R}\))
Substituting the given values, we have:
\[T - (0.0053 \cdot 9.8) = \left(\frac{1}{2} \cdot 0.00397 \cdot 5.0^2\right) \cdot \left(\frac{4.19}{5.0}\right)\]
Simplifying the equation:
\[T - 0.05194 = 0.0248225 \cdot 0.838\]
\[T - 0.05194 \approx 0.0207836\]
\[T \approx 0.05194 + 0.0207836\]
\[T \approx 0.0727236 \, \text{N}\]
The tension in the string is approximately \(0.0727 \, \text{N}\).
To calculate the distance the object rotates downwards in 3.2 seconds, we need to know the initial angular velocity or the angular displacement of the object. Without this information, we cannot provide an accurate calculation for the distance traveled in 3.2 seconds.