Problem 9 - Classical Mechanics

Question

An orbiting space station may be made to rotate to provide an artificial gravity. Given that the radius is \(25 \mathrm{~m}\), find the rotation period required to produce an apparent gravity equal to \(0.7 g\). A man whose normal weight is \(75 \mathrm{~kg}\) weight runs around the station in one direction and then the other (i.e., on a circle on the inside of the cylindrical wall) at \(5 \mathrm{~ms}^{-1}\). Find his apparent weight in each case. What effects will he experience if he climbs a ladder to a higher level ( i.e., closer to the axis), climbing at \(1 \mathrm{~ms}^{-1} ?\)

Step-by-Step Solution

Answer

#Answer# - The rotation period required for the space station to produce an artificial gravity of 0.7g is approximately 15.92 seconds. - When the man runs in the same direction as the rotation, his apparent weight remains the same at 514.725 N. - When the man runs in the opposite direction, his apparent weight decreases to 16.5 N. - When the man climbs the ladder, there will be no net upward or downward force. However, he will experience a lower apparent weight because the centripetal acceleration decreases as he gets closer to the axis of the space station.

Step 1: Find the rotation period

To find the rotation period required for the space station, we can use the formula for artificial gravity: \(F = m \cdot a_{c}\), where \(F\) is the force, \(m\) is the mass (75 kg), and \(a_c\) is the centripetal acceleration. We have \(a_c = 0.7g\) and we know that \(g = 9.81 \mathrm{~m/s^2}\). Moreover, we can relate the centripetal acceleration with the angular velocity \(\omega\) and the radius \(R\): \(a_c = R \cdot \omega^2 \). Finally, the angular velocity can be expressed as \(\omega = \frac{2\pi}{T}\), where \(T\) is the rotation period. Now, we will find \(T\).

Step 2: Relate centripetal acceleration and angular velocity

First, let's relate the centripetal acceleration with the angular velocity: \(a_c = R \cdot \omega^2 \Rightarrow 0.7g = R \cdot \left(\frac{2\pi}{T}\right)^2 \Rightarrow 0.7(9.81) = 25 \cdot \frac{4 \pi^2}{T^2}\).

Step 3: Compute the rotation period, T

Solving for \(T\): \(T^2 = \frac{25 \cdot 4 \pi^2}{0.7(9.81)} \Rightarrow T \approx 15.92 \mathrm{s}\). So, the rotation period required is approximately 15.92 seconds.

Step 4: Find the apparent weights of the man running in each direction

Since the man runs at a speed of \(5 \mathrm{~m/s}\), we can consider two different scenarios: running in the same direction as the rotation, and running in the opposite direction. When running in the same direction, the effective speed of the man is the sum of the speeds \(v_{total} = v_{station} + v_{man}\), where \(v_{station} = R\omega\). The centripetal acceleration stays the same, so his effective weight stays the same as well. When running in the opposite direction, the effective speed is the difference between the speeds \(v_{total} = |v_{station} - v_{man}|\). The centripetal acceleration will be \(a_c' = \frac{v_{total}^2}{R}\), and his apparent weight will be: \(W_{apparent} = m \cdot a_c'\).

Step 5: Compute the apparent weights in each case

Calculate the apparent weights in both scenarios: Running in the same direction: \(W_{same} = m \cdot a_c = 75 \cdot 0.7(9.81) \approx 514.725 \mathrm{N}\). Running in the opposite direction: First, compute \(v_{total}\): \(v_{total} = |v_{station} - v_{man}| = |25\cdot \frac{2\pi}{15.92} - 5| \approx 2.34 \mathrm{~m/s}\), Then, calculate the centripetal acceleration: \( a_c' = \frac{v_{total}^2}{R} \approx 0.22 \mathrm{~m/s^2}\), Finally, compute the apparent weight: \(W_{opposite} = m \cdot a_c' \approx 16.5 \mathrm{N}\). So, the apparent weights are approximately 514.725 N in the same direction and 16.5 N in the opposite direction.

Step 6: Analyze the effects when climbing the ladder

When the man climbs the ladder at \(1\mathrm{~m/s}\), we need to compute the effective force acting on him in the radial direction. As he gets closer to the axis, the value of R decreases, which means the centripetal acceleration also decreases. Thus, the net upward or downward force will depend on the difference between the gravitational force and the centripetal acceleration. Since he climbs at a constant speed, there will be no net upward or downward force. However, he will experience a lower apparent weight as he gets closer to the axis, because the centripetal acceleration will decrease.

Key Concepts

Centripetal Acceleration

Centripetal acceleration is a crucial concept in understanding artificial gravity in rotating systems, like a space station. It occurs when an object is moving along a circular path and is directed towards the center of the circle.
For example, in the case of a rotating space station, the force that provides the necessary centripetal acceleration is what causes the sensation of "gravity" inside the station.
This means that the centripetal force is acting on every object inside, pulling them toward the station's center, but because of the rotation, it feels like a downward force similar to gravity.To calculate centripetal acceleration, you can use the formula:

• \(a_c = \frac{v^2}{R}\), where \(v\) is the velocity and \(R\) is the radius of the circular path.
• Alternatively, it's also \(a_c = R \cdot \omega^2\), with \(\omega\) being the angular velocity.

Understanding this concept is essential for calculating other elements like the apparent weight and the period of rotation required for specific levels of artificial gravity.

Rotation Period

The rotation period is an essential aspect of creating artificial gravity in a rotating space station. It refers to the time taken for the station to complete one full rotation.
By adjusting the rotation period, the level of artificial gravity inside the station can be controlled.
To achieve the desired artificial gravity, we need to relate the rotation period to centripetal acceleration and angular velocity.The relationship is expressed using the following equations:

• Start with \(a_c = 0.7g\), where \(g = 9.81 \mathrm{~m/s^2}\) is the standard gravity.
• Relate it to angular velocity: \( a_c = R \cdot \left(\frac{2\pi}{T}\right)^2\)
• Solve for rotation period \(T\) using: \(T = \sqrt{\frac{25 \cdot 4 \pi^2}{0.7(9.81)}}\)

By solving these equations, we find the period needed for the station's rotation to simulate the desired level of gravity. In our exercise case, this came out to be approximately 15.92 seconds for each full rotation.

Apparent Weight

Apparent weight is another key concept in rotating environments. It refers to the force an object feels as a result of the centripetal acceleration rather than the usual gravitational force.
In the space station, this apparent weight changes depending on factors like the direction one is moving relative to the rotation. The apparent weight is computed by adjusting the centripetal acceleration, considering whether a person is moving along with or against the rotation:

• When moving in the same direction as the station's rotation, the apparent weight remains higher.
• In the opposite direction, it significantly decreases due to reduced effective velocity, resulting in lower centripetal acceleration.

If a person climbs toward the center, their apparent weight reduces, because as they move closer to the axis, the force acting on them diminishes, lowering the sensation of gravity.
Understanding apparent weight helps in predicting experiences and designing exercises for inhabitants of such a rotating spacecraft.