Consider a ball that rolls smoothly down a slope. The ball is released from rest at the top of the slope. The slope has a horizontal end section with a certain height above the ground. After the ball leaves this horizontal section, it lands at a horizontal distance d, away from the end of the slope. Now we apply a perfect lubricant to make the surface of the slope frictionless and then let a block slide down the slope. The block is released from rest at the top of the slope and lands at a horizontal distance d, away from the end of the slope. The ball and the block have the same mass. Find d,Id,.
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- In the figure, a small block of mass m = 0.021 kg can slide along the frictionless loop-the-loop, with loop radius R = 13 cm. The block is released from rest at point P, at height h = 5R above the bottom of the loop. What are the magnitudes of (a) the horizontal component and (b) the vertical component of the net force acting on the block at point Q? (c) At what height h should the block be released from rest so that it is on the verge of losing contact with the track at the top of the loop? (On the verge of losing contact means that the normal force on the block from the track has just then become zero). P h 2T R YA cord is wrapped around a pulley that is shaped like a disk of mass m and radius r. The cord's free end is connected to a block of mass M. The block starts from rest and then slides down an incline that makes an angle O with the horizontal as shown in the figure below. The coefficient of kinetic friction between the block and the incline is µu. (a) Use the concepts of energy to find an expression for the block's speed as a function of position, d, down the incline. (b) Find the magnitude of the acceleration of the block in terms of µ, m, M, g and O. m,r, MJohn wanted to measure the objects angular speed at the bottom of the hill. He built the inclined frictionless ramp of height d, release the object from the top, and measure the angular speed at the Bottom. First, John released a hollow sphere [Ip(2/3) mr^2] from the rest. The diameter and the mass of the sphere were 317 cm and 45.2 kg, respectively. How much was the angular speed John measures at the bottom of the incline he built. Tip: the object rotates down the incline without slipping. Use following values: d=3.57 m and g=9.80 m/s^2
- NZM.11 An unpowered roller-coaster car starts at rest at the top of a hill of height H, rolls down the hill, and then goes around a vertical loop of radius R. Determine the minimum value for H required if the car is to stay on the track at the top of the loop. (Hints: At the top of the loop, the car is upside down. If it is in contact with the track, though, the contact interaction will exert a normal force on the car perpendicular to the track and away from the track, since the normal force is a compression force. You may find it helpful to use conservation of energy here.)Nonuniform ball. a ball of mass M and radius R rolls smoothly from rest down a ramp and onto a circular loop of radius 0.48 m. The initial height of the ball is h =0.36 m. At the loop bottom, the magnitude of the normal force on the ball is 2.00Mg. The ball consists of an outer spherical shell (of a certain uniform density) that is glued to a central sphere (of a different uniform density). The rotational inertia of the ball can be expressed in the general form I = bMR2, but b is not 0.4 as it is for a ball of uniform density. Determine b.In the figure here, a small, solid, uniform ball is to be shot from point P so that it rolls smoothly along a horizontal path, up along a ramp, and onto a plateau. Then it leaves the plateau horizontally to land on a game board, at a horizontal distanced from the right edge of the plateau. The vertical heights are h, = 3.5 cm and h2 = 1.70 cm. With what speed must the ball be shot at point P for it to land at d = 4.5 cm? Ball
- Question in pic.The figure shows a ball with mass m = 2.0 kg attached to the end of a thin rod with length L = 0.79 m and negligible mass. The other end of the rod is pivoted so that the ball can move in a vertical circle. (a) What initial speed must be given the ball so that it reaches the vertically upward position with zero speed? What then is its speed at (b) the lowest point and (c) the point on the right at which the ball is level with the initial point? (d) If the ball's mass were doubled, what would the answer to (a) be?A block of mass m is moving with speed v along a horizontal surface when it collides with a uniform rod of mass 2m and length L attached at one end to a pivot. The surface and pivot have negligible friction. The rod is vertical when the block collides with the end of the rod. The block sticks to the rod, and the block-rod system rotates so that the end of the rod reaches a height h, as shown above. The total rotational inertia of the rod about the pivot is 2mL/3. Express answers in parts (a), (b), and (c) in terms of m, L, v, and physical constants as appropriate. c. Derive an expression for the mechanical energy dissipated during the collision.
- A solid brass ball of mass 22 g and radius 8.51 mm rolls smoothly along a loop-the-loop track when released from rest from a tall ramp leading to the loop-the-loop. The circular loop has a 17 cm radius. (a) What is the minimum height from which you can release the ball so that it will go around the loop-the-loop without falling off? For part (b), assume that the ball is released from rest at a height of 61.83 cm above the bottom of the loop. (b) What is the magnitude of the horizontal force on ball when it is at a height 17 cm going up the hoop? (c)What is the direction of the net force on the ball at the top of the hoop? Î =Energy of a Bullet Dissipated by Plywood. As part of a criminal investigation, you need to determine how much of a bullet's energy is dissipated by a 0.500-inch piece of plywood. You construct a device that consists of three disks that are separated by a distance d = 0.850 m and rotate on a common axis. The bullet is fired through the first disk (a few inches above its center), which is composed of a light plastic that has a negligible effect on the speed of the bullet. The bullet then passes through the second disk, which is composed of 0.500-inch plywood. Finally, the bullet strikes the third disk, where it becomes embedded. The disks rotate with an angular velocity of w = 86.0 rad/s. The angular displacement between holes in the first and second disks is A012 = 0.269 rad, and the angular displacement between the holes in the second and third disks is A023 = 0.286 rad. If the mass of the bullet is 12.0 g, find (a) the initial speed of the bullet and (b) the energy dissipated by the…A thin cylindrical ring starts from rest at a height h1 = 98 m. The ring has a radius R = 34 cm and a mass M = 4 kg. Part (a) Write an expression for the ring's initial energy at point 1, assuming that the gravitational potential energy at point 3 is zero. Part (b) If the ring rolls (without slipping) all the way to point 2, what is the ring's energy at point 2 in terms of h2 and v2? Part (c) Given h2 = 32 m, what is the velocity of the ring at point 2 in m/s? Part (d) What is the ring's rotational velocity in rad/s? Part (e) After passing point 2 the hill becomes frictionless and the ring's rotational velocity remains constant. What is the linear velocity of the ring at point 3 in m/s?