Physics 11 Fall 2012 Practice Problems 6
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1 Physics 11 Fall 2012 Practice Problems 6 1. Two points are on a disk that is turning about a fixed axis perpendicular to the disk and through its center at increasing angular velocity. One point is on the rim and the other point is halfway between the rim and the center. (a) Which point moves the greater distance in a given time? (b) Which point turns through the greater angle? (c) Which point has the greater speed? (d) Which point has the greater angular speed? (e) Which point has the greater tangential acceleration? (f) Which point has the greater angular acceleration? (g) Which point has the greater centripetal acceleration? 1
2 2. What is the angular speed of Earth in radians per second as it rotates about its axis? 2
3 3. The methane molecule (CH 4 ) has four hydrogen atoms located at the vertices of a regular tetrahedron of edge length 0.18 nm, with the carbon atom at the center of the tetrahedron. Find the moment of inertia of this molecule for rotation about an axis that passes through the centers of the carbon atom and one of the hydrogen atoms. 3
4 4. During most of its lifetime, a star maintains an equilibrium size in which the inward force of gravity on each atom is balanced by an outward pressure force due to the heat of the nuclear reactions in core. But after all the hydrogen fuel is consumed by nuclear fusion, the pressure force drops and the star undergoes gravitational collapse until it becomes a neutron star. In a neutron star, the electrons and protons are squeezed together by gravity until they form neutrons. Neutron stars spin very rapidly and emit intense pulses of radio and light waves, one pulse per rotation. These pulsing stars were discovered in the 1960s and are called pulsars. (a) A star with the mass (M = kg) and size (R = m) of our sun rotates once every 30 days. After undergoing gravitational collapse, the star forms a pulsar that is observed by astronomers to emit radio pulses every 0.10 seconds. By treating the neutron star as a solid sphere, deduce its radius. (b) What is the speed of a point on the equator of the neutron star? Your answer will be somewhat too large because a star cannot be accurately modeled as a solid sphere. Even so, you will be able to show that a star, whose mass is 10 6 larger than the earth s, can be compressed by gravitational forces to a size smaller than a typical state in the United States! 4
5 5. A pendulum consisting of a string of length L attached to a bob of mass m swings in a vertical plane. When the string is at an angle θ to the vertical, (a) calculate the tangential acceleration of the bob using F t = ma t. (b) What is the torque exerted about the pivot point? (c) Show that τ = Iα with a t = Lα gives the same tangential acceleration as found in Part (a). 5
6 6. A uniform 1.5 m diameter ring is pivoted at a point on its perimeter so that it is free to rotate about a horizontal axis that is perpendicular to the plane of the ring. The ring is released with the center of the ring at the same height as the axis. (a) If the ring was released from rest, what was its maximum angular speed? (b) What minimum angular speed must it be given at release if it is to rotate a full 360? 6
7 7. A uniform solid sphere of mass M and radius R is free rotate about a horizontal axis through its center. A string is wrapped around the sphere and is attached to an object of mass m. Assume that the string does not slip on the sphere. Find (a) the acceleration of the object and (b) the tension in the string. 7
8 8. A basketball rolls without slipping down an incline of angle θ. The coefficient of static friction is µ s. Model the ball as a thin spherical shell. Find (a) the acceleration of the center of mass of the ball, (b) the frictional force acting on the ball, and (c) the maximum angle of the incline for which the ball will roll without slipping. 8
9 9. Released from rest at the same height, a thin spherical shell and a solid sphere of the same mass m and radius R roll without slipping down an incline through the same vertical drop H. Each is moving horizontally as it leaves the ramp. The spherical shell hits the ground a horizontal distance L from the end of the ramp and the solid sphere hits the ground a distance L from the end of the ramp. Find the ratio L /L. 9
10 10. According to the Standard Model of Particle Physics, electrons are pointlike particles having no spatial extent. (This assumption has been confirmed experimentally, and the radius of the electron has been shown to be less than meters.) The intrinsic spin of an electron could in principle be due to its rotation. Let us check to see if this conclusion is feasible. (a) Assuming that the electron is a uniform sphere whose radius is m, what angular speed would be necessary to produce the observed intrinsic angular momentum of /2? (b) Using this value of the angular speed, show that the speed of a point on the equator of a spinning electron would be moving faster than the speed of light. What is your conclusion about the spin angular momentum being analogous to a spinning sphere with spatial extent? 10
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