Questions: What is the tension in the lower cord? Express your answer with the appropriate units. What is the speed of the block? Express your answer with the appropriate units.

Solution Steps

We need to determine the tension in the lower cord and the speed of the block in a rotating system. The block has a mass of 4.00 kg and is attached to a vertical rod by two strings. The system rotates about the axis of the rod.

When the system rotates, the block experiences centripetal force. The forces acting on the block are:

1. Tension in the upper string (\(T_1\))
2. Tension in the lower string (\(T_2\))
3. Gravitational force (\(mg\))

The block is in equilibrium in the vertical direction, so the sum of the vertical components of the tensions must equal the gravitational force: \[ T_1 \cos(\theta) + T_2 \cos(\theta) = mg \]

In the horizontal direction, the centripetal force is provided by the horizontal components of the tensions: \[ T_1 \sin(\theta) + T_2 \sin(\theta) = m \frac{v^2}{r} \]

Given that the tension in the upper string is known, we can solve for \(T_2\). However, the problem does not provide the value of \(T_1\) or \(\theta\). Assuming we have these values, we can solve the vertical equilibrium equation for \(T_2\): \[ T_2 = \frac{mg - T_1 \cos(\theta)}{\cos(\theta)} \]

Using the horizontal force balance equation, we can solve for the speed \(v\): \[ v = \sqrt{\frac{r (T_1 \sin(\theta) + T_2 \sin(\theta))}{m}} \]

Final Answer