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.

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 eT

A 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…

A 59 kg skier leaves the end of a ski-jump ramp with a velocity of 26 m/s directed 20° above the horizontal. Suppose that as a result of air drag the skier returns to the ground with a speed of 17 m/s, landing 11 m vertically below the end of the ramp. From the launch to the return to the ground, by how much is the mechanical energy of the skier-Earth system reduced because of air drag?

A 50.6 g golf ball is driven from the tee with an initial speed of 55.4 m/s and roses to a height of 23.8 m (a) Neglect air resistance and determine the kinetic energy of the ball at its highest point. (b) what is it speed when it is 5.63 m below its highest point ?