Two thin rods are fastened to the inside of a circular ring as shown in Figure P2.42. One rod of length D is vertical, and the other of length L makes an angle θ with the horizontal. The two rods and the ring lie in a vertical plane. Two small beads are free to slide without friction along the rods. (a) If the two beads are released from rest simultaneously from the positions shown, use your intuition and guess which bead reaches the bottom first. (b) Find an expression for the time interval required for the red head to fall from point Ⓐ to point Ⓒ in terms of g and D . (c) Find an expression for the time interval required for the blue bead to slide from point Ⓑ to point Ⓒ in terms of g , L , and θ . (d) Show that the two time intervals found in parts (b) and (c) are equal. Hint: What is the angle between the chords of the circle Ⓐ Ⓑ and Ⓑ Ⓒ? (e) Do these results surprise you? Was your intuitive guess in part (a) correct? This problem was inspired by an article by Thomas B. Greenslade, Jr., “Galileo’s Paradox,” Phys . Teach . 46 , 294 (May 2008). Figure P2.42
Two thin rods are fastened to the inside of a circular ring as shown in Figure P2.42. One rod of length D is vertical, and the other of length L makes an angle θ with the horizontal. The two rods and the ring lie in a vertical plane. Two small beads are free to slide without friction along the rods. (a) If the two beads are released from rest simultaneously from the positions shown, use your intuition and guess which bead reaches the bottom first. (b) Find an expression for the time interval required for the red head to fall from point Ⓐ to point Ⓒ in terms of g and D . (c) Find an expression for the time interval required for the blue bead to slide from point Ⓑ to point Ⓒ in terms of g , L , and θ . (d) Show that the two time intervals found in parts (b) and (c) are equal. Hint: What is the angle between the chords of the circle Ⓐ Ⓑ and Ⓑ Ⓒ? (e) Do these results surprise you? Was your intuitive guess in part (a) correct? This problem was inspired by an article by Thomas B. Greenslade, Jr., “Galileo’s Paradox,” Phys . Teach . 46 , 294 (May 2008). Figure P2.42
Solution Summary: The author analyzes the first guess on which bead reaches the bottom first. The blue travels a shorter distance with an acceleration of gmathrm
Two thin rods are fastened to the inside of a circular ring as shown in Figure P2.42. One rod of length D is vertical, and the other of length L makes an angle θ with the horizontal. The two rods and the ring lie in a vertical plane. Two small beads are free to slide without friction along the rods. (a) If the two beads are released from rest simultaneously from the positions shown, use your intuition and guess which bead reaches the bottom first. (b) Find an expression for the time interval required for the red head to fall from point Ⓐ to point Ⓒ in terms of g and D. (c) Find an expression for the time interval required for the blue bead to slide from point Ⓑ to point Ⓒ in terms of g, L, and θ. (d) Show that the two time intervals found in parts (b) and (c) are equal. Hint: What is the angle between the chords of the circle Ⓐ Ⓑ and Ⓑ Ⓒ? (e) Do these results surprise you? Was your intuitive guess in part (a) correct? This problem was inspired by an article by Thomas B. Greenslade, Jr., “Galileo’s Paradox,” Phys. Teach. 46, 294 (May 2008).
air is pushed steadily though a forced air pipe at a steady speed of 4.0 m/s. the pipe measures 56 cm by 22 cm. how fast will air move though a narrower portion of the pipe that is also rectangular and measures 32 cm by 22 cm
No chatgpt pls will upvote
13.87 ... Interplanetary Navigation. The most efficient way
to send a spacecraft from the earth to another planet is by using a
Hohmann transfer orbit (Fig. P13.87). If the orbits of the departure
and destination planets are circular, the Hohmann transfer orbit is an
elliptical orbit whose perihelion and aphelion are tangent to the
orbits of the two planets. The rockets are fired briefly at the depar-
ture planet to put the spacecraft into the transfer orbit; the spacecraft
then coasts until it reaches the destination planet. The rockets are
then fired again to put the spacecraft into the same orbit about the
sun as the destination planet. (a) For a flight from earth to Mars, in
what direction must the rockets be fired at the earth and at Mars: in
the direction of motion, or opposite the direction of motion? What
about for a flight from Mars to the earth? (b) How long does a one-
way trip from the the earth to Mars take, between the firings of the
rockets? (c) To reach Mars from the…
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