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Rolling down an Incline Consider a solid uniform disk that starts from rest and rolls without slipping down a ramp sloping at an angle θ, as shown in Figure 9-10(a). Derive expressions for the acceleration of the centre of mass of the disk and for the force of friction between the surface of the ramp and the disk.
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Electronic Engineering
Kinetic Energy of a Rolling Wheel A spoked wheel of mass m and radius r rolls without slipping. Most of the mass of the wheel is distributed along its rim, so the wheel can be treated as a hoop or a ring. Derive an expression for the total kinetic energy of the wheel when the speed of the centre of mass is v_cm. Starting from rest, the wheel rolls without slipping down a ramp from a height h. Derive an expression for the speed of the centre of mass of the wheel at the bottom of the ramp.
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The Speed of a Sphere Rolling Downhill Starting from rest, a solid uniform sphere with mass m and radius r rolls without slipping down a ramp with height h (Figure 9-16). Derive an expression for the speed of the sphere at the bottom of the ramp.
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The Acceleration of a Sphere Rolling down a Ramp Using Work and Energy Using energy considerations, derive an expression for the linear acceleration of an object of mass m, moment of inertia I, and radius r as it rolls without slipping down a ramp with a slope of angleθ. Apply your results to a solid uniform sphere and to a solid uniform disk. Compare your results for the disk to the results in Example 9-2, which was done using Newton’s second law.
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Acceleration in a Pulley System Figure 9-17 shows a rope that is wound around a solid cylinder of mass m_c and radius R. The free end of the rope passes over a pulley of radius r and moment of inertia I^P and is attached to a hanging mass M. The system is initially at rest. Mass M is then allowed to fall, pulling the rope and causing the pulley to rotate and the cylinder to roll as the rope unwinds from the cylinder. The cylinder rolls without slipping. Derive an expression for the acceleration of the hanging mass.
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Ball on a Vertical Circular track The solid metal ball in Figure 9-19 rolls down the ramp, as shown, and then up the vertical circular track. Find the minimum height, h, for which the ball will go all the way to the highest point of the circular track. Assume perfect rolling throughout. The radius of the track is R, and the radius of the ball is r.
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Marble on an Inverted Bowl A bowl of radius R is sitting upside down on a horizontal surface (Figure 9-21). A child holds a marble of radius r at the top of the outside surface of the bowl. The child lets go of the marble, and it rolls without slipping. Derive an expression for the location at which the marble will fly off the surface of the bowl. Assume that the marble will continue rolling and not skid until it loses contact with the surface, which is tantamount to saying the coefficient of static friction between the marble and the bowl is very large. This assumption is put in place so you can comment on the difference between this example and Example 6-14, “Sliding on a Spherical Surface,” in Chapter 6. In this example, we assume that the marble rolls without slipping until it leaves the surface. Is this realistic? Comment on how you expect the marble to behave realistically before it leaves the surface. Compare this example to problems 9-82 and 9-91.
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Rolling with Zero Friction The wheel in Figure 9-24 is an ideal ring/hoop of mass m and radius R, originally at rest. The rope is wound around the wheel a few times so you can cause the wheel to roll by pulling on the rope. Assume the surface provides enough friction to ensure rolling without slipping as needed. Calculate the force of friction on the hoop as you pull the rope horizontally with force F ⃗ ; you may calculate the force of friction in terms of F ⃗.
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Electronic Engineering
