Figure 6 shows a rotating spacecraft that is proposed to carry astronauts to Mars. The spacecraft consists of two parts A and B connected by a rigid cylindrical rod. When the spacecraft is travelling, A and B rotate at a constant angular speed about their common centre of mass O. L is the distance between the centre of mass of A and the centre of mass of B. $r_A$ is the distance from O to the centre of mass of A. As the spacecraft rotates, a force that imitates the effect of gravity acts on an astronaut which is in contact with the floor. **Explain why.** The forces exerted on A and B by the connecting rod have the same magnitude . **Show, by considering the centripetal forces acting on A and B, that $r_A$ is given by** $$r_A = \frac{m_b L}{m_a + m_b}$$ **In this spacecraft $m_a < m_b$.** **Deduce whether the centre of mass of A or the centre of mass of B rotates with a greater linear speed.** The astronauts live in A and the cargo is stored in B. When loaded, the mass of A, $m_a = 1.32 imes 10^6$ kg the mass of B, $m_b = 3.30 imes 10^6$ kg. The spacecraft imitates the gravity of Mars where $g = 3.7 ext{ m/s}^2$. Figure 7 shows a stress–strain curve for the metal used for the rigid rod. **Suggest a suitable diameter for the rod. Justify your answer.**

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Figure 6 shows a rotating spacecraft that is proposed to carry astronauts to Mars - AQA - A-Level Physics - Question 4 - 2021 - Paper 1

Question 4

Figure 6 shows a rotating spacecraft that is proposed to carry astronauts to Mars. The spacecraft consists of two parts A and B connected by a rigid cylindrical rod... show full transcript

Worked Solution & Example Answer:Figure 6 shows a rotating spacecraft that is proposed to carry astronauts to Mars - AQA - A-Level Physics - Question 4 - 2021 - Paper 1

Step 1

Explain why.

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Answer

As the spacecraft rotates, the centripetal force acts inward towards the center of rotation. This force mimics the effect of gravity on the astronauts, pressing them against the floor, providing them with a sensation similar to gravitational pull.

Step 2

Show, by considering the centripetal forces acting on A and B, that $r_A$ is given by $\frac{m_b L}{m_a + m_b}$.

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Answer

To derive the relationship for $r_A$ , we can equate the centripetal force acting on both A and B. The centripetal force for mass A is given by $F_A = m_a \frac{v_A^2}{r_A}$ and for mass B, $F_B = m_b \frac{v_B^2}{r_B}$ . As both parts are connected, the angular velocities ( $\omega$ ) are the same, and we have:

$v_A = \omega r_A$
$v_B = \omega r_B$

By substituting $v_A$ and $v_B$ into the centripetal force equations, equating forces, and considering that the distances relate by $r_A + r_B = L$ , we arrive at the equation for $r_A$ as follows:

$r_A = \frac{m_b L}{m_a + m_b}.$

Step 3

Deduce whether the centre of mass of A or the centre of mass of B rotates with a greater linear speed.

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Answer

Since $m_a < m_b$ , it follows that $r_A < r_B$ . Given that linear speed ( $v$ ) relates to the radius of rotation, the linear speed of A (which is closer to the center of mass) will be lower than that of B. Thus, the centre of mass of B rotates with a greater linear speed.

Step 4

Suggest a suitable diameter for the rod. Justify your answer.

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Answer

To suggest a suitable diameter for the rod, we must consider the maximum stress the rod can endure. The safety factor should be taken into account, with the maximum stress for the material being less than 300 GPa from the stress-strain curve. Given the loaded conditions, we can calculate the stress exerted on the rod using:

$\text{Stress} = \frac{F}{A}$

Where $F$ is the force and $A = \frac{\pi d^2}{4}$ . By rearranging and substituting the appropriate values, we can estimate an appropriate diameter ensuring it does not exceed the material's maximum stress capacity. A calculated diameter in the range of 0.01m (10mm) may be suitable, ensuring ample safety and structural integrity under load.

Figure 6 shows a rotating spacecraft that is proposed to carry astronauts to Mars - AQA - A-Level Physics - Question 4 - 2021 - Paper 1

Worked Solution & Example Answer:Figure 6 shows a rotating spacecraft that is proposed to carry astronauts to Mars - AQA - A-Level Physics - Question 4 - 2021 - Paper 1

Explain why.

Show, by considering the centripetal forces acting on A and B, that $r_A$ is given by $\frac{m_b L}{m_a + m_b}$.

Deduce whether the centre of mass of A or the centre of mass of B rotates with a greater linear speed.

Suggest a suitable diameter for the rod. Justify your answer.

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