A small ball is tied to a string and set rotating with negligible friction in a vertical circle. Prove that the tension in the string at the bottom of the circle exceeds that at the top of the circle by eight times the weight of the ball. Assume the ball's speed is zero as it sails over the top of the circle and there is no additional energy added to the ball during rotation.
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- During a rockslide, a 440 kg rock slides from rest down a hillside that is 500 m long and 300 m high. The coefficient of kinetic friction between the rock and the hill surface is 0.32. (a) If the gravitational potential energy U of the rock-Earth system is set to zero at the bottom of the hilI, what is the value of U just before the slide? (b) How much energy is transferred to thermal energy during the slide? (c) What is the kinetic energy of the rock as it reaches the bottom of the hill?During a rockslide, a 620 kg rock slides from rest down a hillside that is 710 m along the slope and 240 m high. The coefficient of kinetic friction between the rock and the hill surface is 0.28. (a) If the gravitational potential energy U of the rock-Earth system is zero at the bottom of the hill, what is the value of U just before the slide? (b) How much energy is transferred to thermal energy during the slide? (c) What is the kinetic energy of the rock as it reaches the bottom of the hill? (d) What is its speed then?A box of 9kg is released from rest on a rough inclined plane and slides to the bottom. The plane is inclined at a 230 angle to the horizontal and it is 5.1 m high. The coefficient of kinetic friction between the box and the plane is 0.1. Use conservation of energy to determine the speed of the block as it reaches the bottom.
- In the figure, a small block of mass m = 0.029 kg can slide along the frictionless loop-the-loop, with loop radius R = 16 cm. The block is released from rest at point P, at height h = 5R above the bottom of the loop. How much work does the gravitational force do on the block as the block travels from point P to (a) point Q and (b) the top of the loop? If the gravitational potential energy of the block-Earth system is taken to be zero at the bottom of the loop, what is that potential energy when the block is (c) at point P, (d) at point Q, and (e) at the top of the loop? (a) Number i Units (b) Number i Units (c) Number i Units (d) Number i Units (e) Number i Units R eTA bead of mass m sits on a frictionless helical wire whose shape is given by x = 4 cos(0), y = 4 sin(0), z = -50. After the bead is released from rest at z = 0 it slides down due to the gravitational force Fg = = -mg k. b) What is the energy before it is released? E = a) Denoting the speed of the bead as u, give the formula for the energy of this bead. E= * m V= Z=0 c) Give the speed of the bead when it has rotated about the z- axis n times: Here frictionless means that there is no loss of energy due to reaction forces or air resistance.Protons and neutrons (together called nucleons) are held together in the nucleus of an atom by a force called the strong force. At very small separations, the strong force between two nucleons is larger than the repulsive electrical force between two protons--hence its name. But the strong force quickly weakens as the distance between the protons increases. A well-established model for the potential energy of two nucleons interacting via the strong force is U=U0[1−e−x/x0] where x is the distance between the centers of the two nucleons, x0 is a constant having the value x0 = 2.0 × 10−15 m, and U0 = 6.0 × 10−11 J. Quantum effects are essential for a proper understanding of nucleons, but let us innocently consider two neutrons as if they were small, hard, electrically neutral spheres of mass 1.67 ×10−27 kg and diameter 1.0 × 10−15 m. Suppose you hold two neutrons 9.0 × 10−15 m apart, measured between their centers, then release them.What is the speed of each neutron as they crash…
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