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Advanced Physics
4. A tidal force is caused by a non-uniform gravitational field such as the one around the Earth. If you are in such a field, different parts of your body experience slightly different gravitational accelera- tions. As a result, your body parts are stretched or squeezed relative to the body center. Suppose you are orbiting around the Earth with your feet pointing toward the center of the Earth at an alti- tude of 340 km above the surface. The gravitational acceleration of the center of your body is given by 9center GMO 2 where M is the mass of the Earth (5.97 × 1024 kg) and r is the distance from the center of the Earth to your body center (don't forget to add the radius of the Earth 6.37 × 106 m to the altitude). (a) Calculate the gravitational acceleration of your body center in m/s². (b) Your feet are a little bit closer to the Earth than your body center so that they are pulled down more strongly. Let's assume you are 2.00 m tall. Then gfeet = GMO (r - 1.00 m)²* Because the radius of the Earth is much greater than your height, the above equation can be approximated by 9feet = GMO r2 1 ≈ 1.00 m) 2 GMO r2 [1+ 2(1.00 m) r 3 On the other hand, your head is slightly farther away from the Earth than your body center is. Thus, ghead = GMO (r + 1.00 m) 2 ≈ GMO r2 · [1- 2(1.00 m) r That means, your feet would fall toward the Earth faster than your body center, and your head would lag behind the center. As a result, you experience a tidal force stretching your body in the radial direction: the feet are pulled toward the Earth and the head is pulled away from the Earth relative to the body center. Notice that the body center, feet and head are all accelerating toward the center of the Earth but at different rates. Calculate the difference between gravitational accelerations of your head and feet, Ag: Ag Ifeet 9head ≈ 4GM (1.00 m) p3 (c) Replace the Earth with a very compact, massive body with 8 times the mass of the Sun. Calculate Ag. Is it going to be a very uncomfortable experience for you? Let's assume that your body can tolerate Ag up to 9.80 m/s².

4. A tidal force is caused by a non-uniform gravitational field such as the one around the Earth. If you are in such a field, different parts of your body experience slightly different gravitational accelera- tions. As a result, your body parts are stretched or squeezed relative to the body center. Suppose you are orbiting around the Earth with your feet pointing toward the center of the Earth at an alti- tude of 340 km above the surface. The gravitational acceleration of the center of your body is given by 9center GMO 2 where M is the mass of the Earth (5.97 × 1024 kg) and r is the distance from the center of the Earth to your body center (don't forget to add the radius of the Earth 6.37 × 106 m to the altitude). (a) Calculate the gravitational acceleration of your body center in m/s². (b) Your feet are a little bit closer to the Earth than your body center so that they are pulled down more strongly. Let's assume you are 2.00 m tall. Then gfeet = GMO (r - 1.00 m)²* Because the radius of the Earth is much greater than your height, the above equation can be approximated by 9feet = GMO r2 1 ≈ 1.00 m) 2 GMO r2 [1+ 2(1.00 m) r 3 On the other hand, your head is slightly farther away from the Earth than your body center is. Thus, ghead = GMO (r + 1.00 m) 2 ≈ GMO r2 · [1- 2(1.00 m) r That means, your feet would fall toward the Earth faster than your body center, and your head would lag behind the center. As a result, you experience a tidal force stretching your body in the radial direction: the feet are pulled toward the Earth and the head is pulled away from the Earth relative to the body center. Notice that the body center, feet and head are all accelerating toward the center of the Earth but at different rates. Calculate the difference between gravitational accelerations of your head and feet, Ag: Ag Ifeet 9head ≈ 4GM (1.00 m) p3 (c) Replace the Earth with a very compact, massive body with 8 times the mass of the Sun. Calculate Ag. Is it going to be a very uncomfortable experience for you? Let's assume that your body can tolerate Ag up to 9.80 m/s².

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4. A tidal force is caused by a non-uniform gravitational field such as the one around the Earth. If you are in such a field, different parts of your body experience slightly different gravitational accelera- tions. As a result, your body parts are stretched or squeezed relative to the body center. Suppose you are orbiting around the Earth with your feet pointing toward the center of the Earth at an alti- tude of 340 km above the surface. The gravitational acceleration of the center of your body is given by 9center GMO 2 where M is the mass of the Earth (5.97 × 1024 kg) and r is the distance from the center of the Earth to your body center (don't forget to add the radius of the Earth 6.37 × 106 m to the altitude). (a) Calculate the gravitational acceleration of your body center in m/s². (b) Your feet are a little bit closer to the Earth than your body center so that they are pulled down more strongly. Let's assume you are 2.00 m tall. Then gfeet = GMO (r - 1.00 m)²* Because the radius of the Earth is much greater than your height, the above equation can be approximated by 9feet = GMO r2 1 ≈ 1.00 m) 2 GMO r2 [1+ 2(1.00 m) r 3
Transcribed Image Text:4. A tidal force is caused by a non-uniform gravitational field such as the one around the Earth. If you are in such a field, different parts of your body experience slightly different gravitational accelera- tions. As a result, your body parts are stretched or squeezed relative to the body center. Suppose you are orbiting around the Earth with your feet pointing toward the center of the Earth at an alti- tude of 340 km above the surface. The gravitational acceleration of the center of your body is given by 9center GMO 2 where M is the mass of the Earth (5.97 × 1024 kg) and r is the distance from the center of the Earth to your body center (don't forget to add the radius of the Earth 6.37 × 106 m to the altitude). (a) Calculate the gravitational acceleration of your body center in m/s². (b) Your feet are a little bit closer to the Earth than your body center so that they are pulled down more strongly. Let's assume you are 2.00 m tall. Then gfeet = GMO (r - 1.00 m)²* Because the radius of the Earth is much greater than your height, the above equation can be approximated by 9feet = GMO r2 1 ≈ 1.00 m) 2 GMO r2 [1+ 2(1.00 m) r 3
On the other hand, your head is slightly farther away from the Earth than your body center is. Thus, ghead = GMO (r + 1.00 m) 2 ≈ GMO r2 · [1- 2(1.00 m) r That means, your feet would fall toward the Earth faster than your body center, and your head would lag behind the center. As a result, you experience a tidal force stretching your body in the radial direction: the feet are pulled toward the Earth and the head is pulled away from the Earth relative to the body center. Notice that the body center, feet and head are all accelerating toward the center of the Earth but at different rates. Calculate the difference between gravitational accelerations of your head and feet, Ag: Ag Ifeet 9head ≈ 4GM (1.00 m) p3 (c) Replace the Earth with a very compact, massive body with 8 times the mass of the Sun. Calculate Ag. Is it going to be a very uncomfortable experience for you? Let's assume that your body can tolerate Ag up to 9.80 m/s².
Transcribed Image Text:On the other hand, your head is slightly farther away from the Earth than your body center is. Thus, ghead = GMO (r + 1.00 m) 2 ≈ GMO r2 · [1- 2(1.00 m) r That means, your feet would fall toward the Earth faster than your body center, and your head would lag behind the center. As a result, you experience a tidal force stretching your body in the radial direction: the feet are pulled toward the Earth and the head is pulled away from the Earth relative to the body center. Notice that the body center, feet and head are all accelerating toward the center of the Earth but at different rates. Calculate the difference between gravitational accelerations of your head and feet, Ag: Ag Ifeet 9head ≈ 4GM (1.00 m) p3 (c) Replace the Earth with a very compact, massive body with 8 times the mass of the Sun. Calculate Ag. Is it going to be a very uncomfortable experience for you? Let's assume that your body can tolerate Ag up to 9.80 m/s².
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