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).
Over the holiday break you have an internship with an ice skating show. An ice skater will start from rest and slide down an ice-covered ramp. At the bottom of the ramp, the skater will glide around an ice-covered loop which is the inside of a vertical circle before emerging out onto the skating rink floor. For a spectacular effect, the circular loop will have a diameter of 30 feet. Your task is to determine the minimum height from the rink floor to the top of the ramp for the skater to make it around the loop. When barely making it around, the skater briefly loses contact with the ice at the top of the loop.
A 110 g ball bobbing up and down on the ocean as the waves roll by has a vertical position that can be described as a function of time by x(t)=cos(10t). What is the force on the ball at a time of 5.4 seconds?
A pendulum has a length l (the rope is massless). The mass of the object suspended from the pendulum is m. With rope horizontal θ = 90o When it makes an angle of degrees, we first leave the object at no speed. Any friction can be neglected. Gravitational acceleration g. Give your answers in terms of l, m and g.
When = 0o, what is the tension in the rope?
University Physics with Modern Physics (14th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.