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pothetical future space station v above. Suppose the space sta - = CMR where M is the giv , and C is a given numerical cc %3D n, the space engineers need to r n and bring the station to rest. T Is shown. Each thruster is attac en distance r from the center and ach thruster will result in a forc

pothetical future space station v above. Suppose the space sta - = CMR where M is the giv , and C is a given numerical cc %3D n, the space engineers need to r n and bring the station to rest. T Is shown. Each thruster is attac en distance r from the center and ach thruster will result in a forc

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F app Thruston Thrustor Ф app Spinning space station orbits Earth I Consider a hypothetical future space station that consists of a torus attached by spokes to a central hub as show above. Suppose the space station has an given initial angular speed of wo and a rotational inertia I = CMR where M is the given mass of the space station, R is the given outer radius of the torus, and C is a given numerical constant. In this situation, the space engineers need to repair the station and in order to do this, they need to stop the rotation and bring the station to rest. Therefore, the engineers affix two large thrusters to the space station as shown. Each thruster is attached to a fixed position on the space station corre- sponding to a given distance r from the center and at a given angle o relative to the radial direction. When activated, each thruster will result in a force of given constant magnitude Fapp applied to the station as shown. Note that a coordinate system is given here, with x-direction corresponding to the right, y-direction corresponding to the top of the page, and z-direction corresponding to out of the page. Part a)- What is the magnitude of the total torque T applied to the space station that results from the action of both of the thrusters? Give your answer in terms of given parameters only. Explain your work. Part b) – What is the magnitude of the angular acceleration a of the space station that results from the action of both of the thrusters? Give your answer in terms of given parameters only. Explain your work. Part c) – What is the linear accelerationā of the entire space station as a single body that results from the action of both of the thrusters? Indicate both the magnitude and the direction. Give your answer in terms of given parameters only. Explain your work. Hint: The Rolling Constraint does not apply here. Part d) - Assume that at time t = 0 both thrusters are activated. How many total rotations of the space station will occur before the space station is brought completely to rest? Give your answer in terms of given parameters only. Explain your work. Hint: one rotation equals 27 radians of angle. z out of page
Transcribed Image Text:F app Thruston Thrustor Ф app Spinning space station orbits Earth I Consider a hypothetical future space station that consists of a torus attached by spokes to a central hub as show above. Suppose the space station has an given initial angular speed of wo and a rotational inertia I = CMR where M is the given mass of the space station, R is the given outer radius of the torus, and C is a given numerical constant. In this situation, the space engineers need to repair the station and in order to do this, they need to stop the rotation and bring the station to rest. Therefore, the engineers affix two large thrusters to the space station as shown. Each thruster is attached to a fixed position on the space station corre- sponding to a given distance r from the center and at a given angle o relative to the radial direction. When activated, each thruster will result in a force of given constant magnitude Fapp applied to the station as shown. Note that a coordinate system is given here, with x-direction corresponding to the right, y-direction corresponding to the top of the page, and z-direction corresponding to out of the page. Part a)- What is the magnitude of the total torque T applied to the space station that results from the action of both of the thrusters? Give your answer in terms of given parameters only. Explain your work. Part b) – What is the magnitude of the angular acceleration a of the space station that results from the action of both of the thrusters? Give your answer in terms of given parameters only. Explain your work. Part c) – What is the linear accelerationā of the entire space station as a single body that results from the action of both of the thrusters? Indicate both the magnitude and the direction. Give your answer in terms of given parameters only. Explain your work. Hint: The Rolling Constraint does not apply here. Part d) - Assume that at time t = 0 both thrusters are activated. How many total rotations of the space station will occur before the space station is brought completely to rest? Give your answer in terms of given parameters only. Explain your work. Hint: one rotation equals 27 radians of angle. z out of page
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