Two skaters, each of mass 35 kg, approach each other along parallel paths separated by 3.6 m. They have equal and opposite velocities of 1.4 m/s. The first skater carries one end of a long pole with negligible mass, and the second skater grabs the other end of it as she passes. See Fig. 12-42. Assume frictionless ice.Figure 12-42(a) Describe quantitatively the motion of the skaters after they have become connected by the pole.1.8 1.8 m (radius)0.78 0.778 rad/s (angular speed)(b) By pulling on the pole, the skaters reduce their separation to 1.0 m. What is their angular speed then?10.08 10.1 rad/s(c) Calculate the kinetic energy of the system in (a) and (b).69.3 68.6 J (energy for system a)889.056 889 J (energy for system b)(d) What is the source of the added kinetic energy?
Question
Two skaters, each of mass 35 kg, approach each other along parallel paths separated by 3.6 m. They have equal and opposite velocities of 1.4 m/s. The first skater carries one end of a long pole with negligible mass, and the second skater grabs the other end of it as she passes. See Fig. 12-42. Assume frictionless ice.Figure 12-42(a) Describe quantitatively the motion of the skaters after they have become connected by the pole.1.8 1.8 m (radius)0.78 0.778 rad/s (angular speed)(b) By pulling on the pole, the skaters reduce their separation to 1.0 m. What is their angular speed then?10.08 10.1 rad/s(c) Calculate the kinetic energy of the system in (a) and (b).69.3 68.6 J (energy for system a)889.056 889 J (energy for system b)(d) What is the source of the added kinetic energy?
Solution
(a) After the skaters grab the pole, they will start rotating around the center of mass of the system. The center of mass is in the middle of the pole (1.8 m from each skater). The angular speed can be calculated using the formula ω = v/r, where v is the linear speed and r is the radius. Substituting the given values, we get ω = 1.4 m/s / 1.8 m = 0.778 rad/s.
(b) When the skaters pull on the pole and reduce their separation to 1.0 m, the radius of the rotation decreases. However, the angular momentum of the system is conserved. Therefore, the angular speed increases. It can be calculated using the formula ω' = ω * r / r', where ω is the initial angular speed, r is the initial radius, and r' is the final radius. Substituting the given values, we get ω' = 0.778 rad/s * 1.8 m / 1.0 m = 10.1 rad/s.
(c) The kinetic energy of the system in both cases can be calculated using the formula K = 1/2 * I * ω^2, where I is the moment of inertia and ω is the angular speed. The moment of inertia for two point masses at a distance r is I = 2 * m * r^2. Substituting the given values, we get K = 1/2 * 2 * 35 kg * (1.8 m)^2 * (0.778 rad/s)^2 = 68.6 J for the first case and K = 1/2 * 2 * 35 kg * (1.0 m)^2 * (10.1 rad/s)^2 = 889 J for the second case.
(d) The added kinetic energy comes from the work done by the skaters when they pull on the pole. This work is transformed into kinetic energy, causing the increase in the system's kinetic energy.
Similar Questions
Two skaters, each of mass 85 kg, approach each other along parallel paths separated by 5.5 m. They have equal and opposite velocities of 1.9 m/s. The first skater carries one end of a long pole with negligible mass, and the second skater grabs the other end of it as she passes; see the figure. Assume frictionless ice. Describe quantitatively the motion of the skaters after they have become connected by the pole. What is their angular speed?
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11 Skaters of masses 80 kg and 40 kg move directly towards each other and collide.Before the collision, the heavier skater is moving to the right at a speed of 2.0 m s –1 and the lighterskater is moving to the left at a speed of 1.0 m s –1.After the collision, the heavier skater moves to the right at a speed of 0.80 m s –1.What is the relative speed of separation of the two skaters?A 0.6 m s –1 B 1.4 m s –1 C 2.2 m s –1 D 2.6 m s –112 Which statement describes the two forces in a couple?A They act in the same direction.B They act through the same point.C They produce zero resultant force.D They produce zero resultant moment
At the centre of a circular ice rink, a 89.0 kg ice skater moving 1.80 m/s north hits and grabs onto a 59.0 kg ice skater who had been traveling west at 5.35 m/s. If the skaters hold onto each other and the rink has a 36.0 m diameter, what is the amount of time for the skaters to reach the edge of the rink?
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