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).
A Space-X shuttle is fired from its launch pad at x = 0 and travels along a path described by equation below. The x-component of the rocket is also described below and where t is in seconds. Find the magnitude of the shuttle's velocity after t = ten seconds.
a.) 1003 m/s
b.) 1102 m/s
c.) 850 m/s
d.) 950 m/s
The leaning tower of Pisa is 56 m tall. Supposedly Galileo investigated the behavior of falling objects by dropping them from the top of this tower and timing their descent. How long would it take an object to fall 56 m if air effects are negligible?
a. 3.33 s
b. 3.38 s
c. 3.83 s
d. 3.88 s
When a projectile reaches the highest point the horizontal component of the acceleration is:
E) No answer
D. Zero
A. Less than g
B. Positive g
C. Negative g
Chapter 2 Solutions
Physics for Scientists and Engineers, Technology Update (No access codes included)
University Physics with Modern Physics (14th Edition)
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