Question27. Astronauts in orbit are apparently weightless. This means that a clever method of measuring the mass of astronauts is needed to monitor their mass gains or losses, and adjust their diet. One way to do this is to exert a known force on an astronaut and measure the acceleration produced. Suppose a net external force of 50.0 N is exerted, and an astronaut's acceleration is measured to be $0.893 \mathrm{~m} / \mathrm{s}^{2}$ . (a) Calculate her mass. (b) By exerting a force on the astronaut, the vehicle in which she orbits experiences an equal and opposite force. Use this knowledge to find an equation for the acceleration of the system (astronaut and spaceship) that would be measured by a nearby observer. (c) Discuss how this would affect the measurement of the astronaut's acceleration. Propose a method by which recoil of the vehicle is avoided.

Studdy Solution

STEP 1

1. Newton's second law of motion applies: $F = ma$ .
2. The force exerted on the astronaut is 50.0 N.
3. The astronaut's acceleration is $0.893 \, \mathrm{m/s}^2$ .
4. The system includes both the astronaut and the spaceship.
5. The force exerted on the astronaut causes an equal and opposite force on the spaceship.

STEP 2

1. Calculate the astronaut's mass using Newton's second law.
2. Derive the equation for the acceleration of the system.
3. Discuss the effect of the vehicle's recoil on the measurement and propose a method to avoid it.

STEP 3

Using Newton's second law, $F = ma$ , where $F = 50.0 \, \mathrm{N}$ and $a = 0.893 \, \mathrm{m/s}^2$ .
Rearrange the equation to solve for mass $m$ : $m = \frac{F}{a}$
Substitute the given values: $m = \frac{50.0 \, \mathrm{N}}{0.893 \, \mathrm{m/s}^2}$
Calculate the mass: $m \approx 56.0 \, \mathrm{kg}$

STEP 4

Consider the system of the astronaut and the spaceship. When a force is exerted on the astronaut, an equal and opposite force is exerted on the spaceship.
Let $M$ be the mass of the spaceship, and $m$ be the mass of the astronaut.
The total force on the system is zero because the forces are equal and opposite.
The acceleration of the system $a_{\text{system}}$ is given by: $a_{\text{system}} = \frac{F_{\text{net}}}{M + m}$
Since $F_{\text{net}} = 0$ , the acceleration $a_{\text{system}} = 0$ .

STEP 5

The recoil of the vehicle affects the measurement of the astronaut's acceleration because it introduces an additional motion that can alter the observed acceleration.
To avoid the recoil of the vehicle, a method such as using a tether or a stationary reference point within the vehicle can be employed. This would allow the force to be applied in a way that minimizes the effect on the vehicle's motion.
The astronaut's mass is approximately $56.0 \, \mathrm{kg}$ , and the system's acceleration is zero due to equal and opposite forces. A method to avoid vehicle recoil is necessary for accurate measurements.

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