Review. A clown balances a small spherical grape at the top of his bald head, which also has the shape of a sphere. After drawing sufficient applause, the grape starts from rest and rolls down without slipping. It will leave contact with the clown’s scalp when the radial line joining it to the center of curvature makes what angle with the vertical?
Hence, the required angle is
Consider the free body diagram shown below for the given situation.
The energy conservation between the apex and the point where the grape leaves meet the surface.
From this, we get
Take a look at the radial forces at work on the grape.
We obtain at the moment where the grape leaves the surface
We get by substituting equation into equation .
As a result, the needed angle is
An elevator system in a tall building consists of a car and a role="math" localid="1663753328337" counterweight joined by a light cable of constant length that passes over a pulley of mass . The pulley, called a sheave, is a solid cylinder of radius turning on a horizontal axle. The cable does not slip on the sheave. A number n of people, each of mass , are riding in the elevator car, moving upward at and approaching the floor where the car should stop. As an energy-conservation measure, a computer disconnects the elevator motor at just the right moment so that the sheave–car– counterweight system then coasts freely without friction and comes to rest at the floor desired. There it is caught by a simple latch rather than by a massive brake. (a) Determine the distance the car coasts upward as a function of . Evaluate the distance for (b) , (c) , and (d) . (e) For what integer values of does the expression in part (a) apply? (f) Explain your answer to part (e). (g) If an infinite number of people could fit on the elevator, what is the value of ?
A 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 with the horizontal as shown in Figure P10.92. The coefficient of kinetic friction between block and incline is m.
(a) Use energy methods to show that the block’s speed as a function of position d down the incline is
(b) Find the magnitude of the acceleration of the block in terms of , , m,M , g and .
Why is the following situation impossible? In a large city with an air-pollution problem, a bus has no combustion engine. It runs over its citywide route on energy drawn from a large, rapidly rotating flywheel under the floor of the bus. The flywheel is spun up to its maximum rotation rate of 3000 rev/min by an electric motor at the bus terminal. Every time the bus speeds up, the flywheel slows down slightly. The bus is equipped with regenerative braking so that the flywheel can speed up when the bus slows down. The flywheel is a uniform solid cylinder with mass 1200 kg and radius 0.500 m. The bus body does work against air resistance and rolling resistance at the average rate of 25.0 hp as it travels its route with an average speed of 35.0 km/h.
A uniform solid disk of radius is free to rotate on a frictionless pivot through a point on its rim (Fig. P10.57). If the disk is released from rest in the position shown by the copper colored circle,
(a) What is the speed of its center of mass when the disk reaches the position indicated by the dashed circle?
(b) What is the speed of the lowest point on the disk in the dashed position?
(c) What If? Repeat part (a) using a uniform hoop.
Big Ben, the nickname for the clock in Elizabeth Tower (named after the Queen in 2012) in London, has an hour hand long with a mass of 60.0 kg and a minute hand 4.50 m long with a mass of 100 kg (Fig. P10.49). Calculate the total rotational kinetic energy of the two hands about the axis of rotation. (You may model the hands as long, thin rods rotated about one end. Assume the hour and minute hands are rotating at a constant rate of one revolution per 12 hours and 60 minutes, respectively.)
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