Figure 6 shows a rotating spacecraft that is proposed to carry astronauts to Mars. Figure 6 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 who is in contact with the floor. Explain why. The forces exerted on A and B by the connecting rod have the same magnitude. $m_a$ is the mass of A $m_b$ is the mass of B Show, by considering the centripetal forces acting on A and B, that $r_A$ is given by $$r_A = rac{m_b L}{m_a + m_b}$$ 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, $m_a = 1.32 imes 10^6$ kg $m_b = 3.30 imes 10^6$ 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. Figure 7 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. Figure 6 The spacecraft consists of two parts A and B connected by a rigid cyli... 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, a centripetal force is required to keep the astronauts moving in a circular path. This force acts inwards, toward the center of rotation, which mimics the effect of gravity. The floor of the spacecraft pushes up on the astronauts, providing the necessary force to simulate the effects of gravity during the rotation.

Step 2

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

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Answer

To analyze the forces on A and B, we use the formula for centripetal force, which can be expressed as:

$F_c = \frac{m v^2}{r}$

For mass A rotating at radius $r_A$ , the centripetal force is given by:

$F_{cA} = m_a \frac{v_A^2}{r_A}$

For mass B, which is further from the center at radius $r_B = L - r_A$ , the centripetal force is:

$F_{cB} = m_b \frac{v_B^2}{r_B}$

Since the forces exerted by the rod on A and B are equal, we equate the two expressions and solve for $r_A$ . After substituting $v_A = r_A \omega$ and $v_B = r_B \omega$ , we can find that these relate back to write $r_A$ in terms of $m_a$ , $m_b$ , and L.

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

Given that $m_a < m_b$ , we can analyze the distances. The linear speed is directly proportional to the radius; hence, as $r_A$ is smaller than $r_B$ , the linear speed of the center of mass of B will be greater than that of A. Therefore, 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, we need to consider the maximum stress the rod will undergo. Assuming a safety factor of 3 (as it can withstand the operational stress), we calculate the maximum allowable stress. Given that the maximum stress from the curve in Figure 7 should not exceed 300 MPa, we can use:

$\sigma_{max} = \frac{F}{A} = \frac{m_a g + m_b g}{\frac{\pi d^2}{4}}$

From this, we can rearrange to solve for the diameter $d$ . After plugging in the values for masses and the force due to gravity, we can derive a suitable diameter that ensures the rod can withstand the applied forces without failing.

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

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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