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SOLUTION Conceptualize The change in the neutron star's motion is similar to that of the skater described earlier in the textbook, but in the reverse direction. As the mass of the star moves closer to the rotation axis, we expect the star to spin ---Select-v. Categorize Let us assume that during the collapse of the stellar core, (1) no external torque acts on it, (2) it remains spherical with the same relative mass distribution, and (3) its mass remains constant. We categorize the star as -Select- v system in terms of angular momentum. We do not know the mass distribution of the star, but we have assumed the distribution is symmetric, so the moment of inertia can be expressed as KMR2, where k is some numerical constant. (From this table, for example, we see thatk- for a solid sphere andk- for a spherical shell.) Analyze (Use the following as necessary: a, T, T, R, Rp, k, and M.) Let's use the symbol T for the period, with T, being the initial period of the star and T, being the period of the neutron star. The star's angular speed is given by o- From the isolated system model for angular momentum, write the following equation for the star: 1,0, - Ipo Use o- 2x/T to rewrite this equation in terms of the initial and final periods: Substitute the moments of inertia in the preceding equation: - kMR kMR, Solve for the final period of the star: T,-( Substitute numerical values to find the final period (in s): Finalize The neutron star does indeed rotate --Select- v after it collapses, as predicted. It moves very fast, in fact, rotating (in rotations per second) times each second. EXERCISE A solid metal sphere of diameter D is spinning in a gravity-free region of space with an angular velocity of o, The sphere is slowly heated until it reaches its melting temperature, at which point flattens into a uniform disk of thickness By what factor is the angular velocity changed? (Give your answer as a factor of o,) Hint

SOLUTION Conceptualize The change in the neutron star's motion is similar to that of the skater described earlier in the textbook, but in the reverse direction. As the mass of the star moves closer to the rotation axis, we expect the star to spin ---Select-v. Categorize Let us assume that during the collapse of the stellar core, (1) no external torque acts on it, (2) it remains spherical with the same relative mass distribution, and (3) its mass remains constant. We categorize the star as -Select- v system in terms of angular momentum. We do not know the mass distribution of the star, but we have assumed the distribution is symmetric, so the moment of inertia can be expressed as KMR2, where k is some numerical constant. (From this table, for example, we see thatk- for a solid sphere andk- for a spherical shell.) Analyze (Use the following as necessary: a, T, T, R, Rp, k, and M.) Let's use the symbol T for the period, with T, being the initial period of the star and T, being the period of the neutron star. The star's angular speed is given by o- From the isolated system model for angular momentum, write the following equation for the star: 1,0, - Ipo Use o- 2x/T to rewrite this equation in terms of the initial and final periods: Substitute the moments of inertia in the preceding equation: - kMR kMR, Solve for the final period of the star: T,-( Substitute numerical values to find the final period (in s): Finalize The neutron star does indeed rotate --Select- v after it collapses, as predicted. It moves very fast, in fact, rotating (in rotations per second) times each second. EXERCISE A solid metal sphere of diameter D is spinning in a gravity-free region of space with an angular velocity of o, The sphere is slowly heated until it reaches its melting temperature, at which point flattens into a uniform disk of thickness By what factor is the angular velocity changed? (Give your answer as a factor of o,) Hint

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Formation of a Neutron Star A star rotates with a period of 33 days about an axis through its center. The period is the time interval required for a point on the star's equator to make one complete revolution around the axis of rotation. After the star undergoes a supernova explosion, the stellar core, which had a radius of 2.0 x 10 km, collapses into a neutron star of radius 6.3 km. Determine the period of rotation of the neutron star. SOLUTION Conceptualize The change in the neutron star's motion is similar to that of the skater described earlier in the textbook, but in the reverse direction. As the mass of the star moves doser to the rotation axis, we expect the star to spin ---Select--v. Categorize Let us assume that during the collapse of the stellar core, (1) no external torque acts on it, (2) it remains spherical with the same relative mass distribution, and (3) its mass remains constant. We categorize the star as ---Select-- v system in terms of angular momentum. We do not know the mass distribution of the star, but we have assumed the distribution is symmetric, so the moment of inertia can be expressed as KMR2, where k is some numerical constant. (From this table, for example, we see that k = for a solid sphere andk - for a spherical shell.) Analyze (Use the following as necessary: w, T, T, R, R, k, and M.) Let's use the symbol T for the period, with T, being the initial period of the star and T, being the period of the neutron star. The star's angular speed is given by o- From the isolated system model for angular momentum, write the following equation for the star: I,0, - Ipop Use o- 2x/T to rewrite this equation in terms of the initial and final periods: Substitute the moments of inertia in the preceding equation: kMR Solve for the final period of the star: T,-( T, Substitute numerical values to find the final period (in s): T, = Finalize The neutron star does indeed rotate ---Select- v after it collapses, as predicted. It moves very fast, in fact, rotating (in rotations per second) times each second. EXERCISE A solid metal sphere of diameter D is spinning in a gravity-free region of space with an angular velocity of w, The sphere is slowly heated until it reaches its melting temperature, at which point flattens into a uniform disk of thickness By what factor is the angular velocity changed? (Give your answer as a factor of o,) Hint
Transcribed Image Text:Formation of a Neutron Star A star rotates with a period of 33 days about an axis through its center. The period is the time interval required for a point on the star's equator to make one complete revolution around the axis of rotation. After the star undergoes a supernova explosion, the stellar core, which had a radius of 2.0 x 10 km, collapses into a neutron star of radius 6.3 km. Determine the period of rotation of the neutron star. SOLUTION Conceptualize The change in the neutron star's motion is similar to that of the skater described earlier in the textbook, but in the reverse direction. As the mass of the star moves doser to the rotation axis, we expect the star to spin ---Select--v. Categorize Let us assume that during the collapse of the stellar core, (1) no external torque acts on it, (2) it remains spherical with the same relative mass distribution, and (3) its mass remains constant. We categorize the star as ---Select-- v system in terms of angular momentum. We do not know the mass distribution of the star, but we have assumed the distribution is symmetric, so the moment of inertia can be expressed as KMR2, where k is some numerical constant. (From this table, for example, we see that k = for a solid sphere andk - for a spherical shell.) Analyze (Use the following as necessary: w, T, T, R, R, k, and M.) Let's use the symbol T for the period, with T, being the initial period of the star and T, being the period of the neutron star. The star's angular speed is given by o- From the isolated system model for angular momentum, write the following equation for the star: I,0, - Ipop Use o- 2x/T to rewrite this equation in terms of the initial and final periods: Substitute the moments of inertia in the preceding equation: kMR Solve for the final period of the star: T,-( T, Substitute numerical values to find the final period (in s): T, = Finalize The neutron star does indeed rotate ---Select- v after it collapses, as predicted. It moves very fast, in fact, rotating (in rotations per second) times each second. EXERCISE A solid metal sphere of diameter D is spinning in a gravity-free region of space with an angular velocity of w, The sphere is slowly heated until it reaches its melting temperature, at which point flattens into a uniform disk of thickness By what factor is the angular velocity changed? (Give your answer as a factor of o,) Hint
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