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A comet of mass \( m_c = 3.2 \times 10^{14} \) \( \text{kg} \) is orbiting a star with mass \( m_s = 1.8 \times 10^{30} \) \( \text{kg} \). The comet’s orbit is elliptical. At its closest point, the comet is a distance \( r_1 = 8.3 \times 10^{10} \) \( \text{m} \) from the star, and at its farthest point, the comet is a distance \( r_2 = 4.9 \times 10^{11} \) \( \text{m} \) from the star. What is the change in the kinetic energy of the comet as it moves along its orbit from distance \( r_2 \) to distance \( r_1 \) from the star?

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Young David experimented with slings before tackling Goliath. He found that he could develop an angular speed of \( 8.0 \) \( \text{rev/s} \) in a sling \( 0.60 \) \( \text{m} \) long. If he increased the length to \( 0.90 \) \( \text{m} \), he could revolve the sling only \( 6.0 \) times per second. (a) Which angular speed gives the greater linear speed? (b) What is the centripetal acceleration at \( 8.0 \) \( \text{rev/s} \)? (c) What is the centripetal acceleration at \( 6.0 \) \( \text{rev/s} \)?

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On a harsh winter day, a 1500 kg vehicle takes a circular banked exit ramp (radius R = 150 m; banking angle of 10 degrees) at a speed of 30 mph, since the speed limit is 35 mph. However, the exit ramp is completely iced up (= frictionless). To make matters worse, a wind is blowing parallel to the ramp in a downward direction. The wind exerts a force of 3000 N. Under these conditions, can the driver continue to follow a safe horizontal circle on the exit ramp and stay below the speed limit? To convert mph into m/s use 1 mi = 1607 m and 1 hr is 3600 s.

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