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6. A small particle is placed inside the semicircular tube of radius R shown in the...
6. (Bonus question.) A small uniform cylinder of radius R rolls without slipping along the inside...
6. (Bonus question.) A small uniform cylinder of radius R rolls without slipping along the inside of a large, fixed cylinder of radius r > R as shown in the figure above. mig (a) Use conservation of energy to show that the period of small oscillations (θ « 1) of the rolling cylinder about the equilibrium position O is equivalent to that of a simple pendulum whose length is (r-R). [Note that the rotation rate w of the cylinder is...
A small sphere of mass m and radius r is placed inside a horizontal hollow cylinders...
A small sphere of mass m and radius r is placed inside a horizontal hollow cylinders of radius Now cylinder rotating with some angular acceleration which slowly increases from zero to certain value There is no slipping between two surfaces during motion:- (A) Maximum angle formed by line joining centre will be 0 =sin 2aR 5g (B) Angular velocity of sphere will be m'= where is angular velocity of cylinder R (C) Sphere will keep rotating at lowest position (D)...
The semicircular wire segment with radius a is carrying current I, and is placed in the...
The semicircular wire segment with radius a is carrying current I, and is placed in the x - y plane as shown. The wire is inside a uniform magnetic field B = Boi. Find the magnitude and the direction of the to- tal magnetic force acting on the wire segment. +x (HINT: Consider an infinitesimal arc segment ds on the wire to start with.)
A particle of mass m moves in a circle of radius R at a constant speed...
A particle of mass m moves in a circle of radius
R at a constant speed v, as shown below. The
motion begins at point Q at time t = 0. Determine
the angular momentum of the particle about the axis perpendicular
to the page through point P as a function of time. (Use
any variable or symbol stated above along with the following as
necessary: t.)
A small particle of mass m is at rest on a horizontal circular platform that is...
A small particle of mass m is at rest on a horizontal circular platform that is free to rotate about a vertical axis through its center. The particle is located at a radius r from the axis, as shown in the figure above. The platform begins to rotate with constant angular acceleration α . Because of friction between the particle and the platform, the particle remains at rest with respect to the platform. When the platform has reached angular speed...
A particle of mass m can slide frictionlessly along a circular ring of radius R that...
A particle of mass m can slide frictionlessly along a circular ring of radius R that is rotating about its vertical diameter at a given (fixed) angular velocity of Ohm, as shown in the figure. Derive the equation of motion for the system.
A sphere of radius R surrounds a particle with charge Q located at its center as...
A sphere of radius R surrounds a particle with charge Q located at its center as shown in figure below. Find the electric flux through a circlular cap of half-angle θ. ) (1-2cos0) 2Eo 2e) (1-2cos θ) COS 2cos θ 2u2 (1 - cos 0) (1-sin θ) 2)
Problem 2 Consider a particle on a sphere of fixed radius r and V = 0...
Problem 2 Consider a particle on a sphere of fixed radius r and V = 0 on the spherical surface. The particle is found in either of the following wavefunctions: Y1,1(0,0) or Y1,-1(0,0). a) What is the energy difference between the two possible wavefunctions? b) Determine the magnitude of angular momentum and its projection onto the z-axis for both wavefunctions. c) Do the energy and the magnitude of angular momentum change when the radius r of the sphere is increased?...
2. It is well-known that living on the inside surface of a cylinder of radius R...
2. It is well-known that living on the inside surface of a cylinder of radius R when rotating at a rate of o about its axis in the outer-space, can simulate the same gravitational g, as living on earth if Ro2-g Show that, inside such a cylinder, when projectiles are shot in any arbitrary direction at low speeds (v<< Ro) and to low altitudes at nearby points (Ar<R), the equation of motion of such trajectories (as observed by a an...
(11 points) A uniform solid sphere of mass m and radius r is placed inside a...
(11 points) A uniform solid sphere of mass m and radius r is placed inside a hemispherical bowl of radius R. The sphere is released from rest at an angle theta and rolls without slipping. (a) (6 points) Using Conservation of Energy, to find an expression for the angular speed of the sphere when it reaches the lowest point of the bowl. (b) (6 points) Find the magnitude of the centripetal acceleration of the center of mass of the sphere...
6. (Bonus question.) A small uniform cylinder of radius R rolls without slipping along the inside of a large, fixed cylinder of radius r > R as shown in the figure above. mig (a) Use conservation of energy to show that the period of small oscillations (θ « 1) of the rolling cylinder about the equilibrium position O is equivalent to that of a simple pendulum whose length is (r-R). [Note that the rotation rate w of the cylinder is...
A small sphere of mass m and radius r is placed inside a horizontal hollow cylinders of radius Now cylinder rotating with some angular acceleration which slowly increases from zero to certain value There is no slipping between two surfaces during motion:- (A) Maximum angle formed by line joining centre will be 0 =sin 2aR 5g (B) Angular velocity of sphere will be m'= where is angular velocity of cylinder R (C) Sphere will keep rotating at lowest position (D)...
The semicircular wire segment with radius a is carrying current I, and is placed in the x - y plane as shown. The wire is inside a uniform magnetic field B = Boi. Find the magnitude and the direction of the to- tal magnetic force acting on the wire segment. +x (HINT: Consider an infinitesimal arc segment ds on the wire to start with.)
A particle of mass m moves in a circle of radius
R at a constant speed v, as shown below. The
motion begins at point Q at time t = 0. Determine
the angular momentum of the particle about the axis perpendicular
to the page through point P as a function of time. (Use
any variable or symbol stated above along with the following as
necessary: t.)
A small particle of mass m is at rest on a horizontal circular platform that is free to rotate about a vertical axis through its center. The particle is located at a radius r from the axis, as shown in the figure above. The platform begins to rotate with constant angular acceleration α . Because of friction between the particle and the platform, the particle remains at rest with respect to the platform. When the platform has reached angular speed...
A particle of mass m can slide frictionlessly along a circular ring of radius R that is rotating about its vertical diameter at a given (fixed) angular velocity of Ohm, as shown in the figure. Derive the equation of motion for the system.
A sphere of radius R surrounds a particle with charge Q located at its center as shown in figure below. Find the electric flux through a circlular cap of half-angle θ. ) (1-2cos0) 2Eo 2e) (1-2cos θ) COS 2cos θ 2u2 (1 - cos 0) (1-sin θ) 2)
Problem 2 Consider a particle on a sphere of fixed radius r and V = 0 on the spherical surface. The particle is found in either of the following wavefunctions: Y1,1(0,0) or Y1,-1(0,0). a) What is the energy difference between the two possible wavefunctions? b) Determine the magnitude of angular momentum and its projection onto the z-axis for both wavefunctions. c) Do the energy and the magnitude of angular momentum change when the radius r of the sphere is increased?...
2. It is well-known that living on the inside surface of a cylinder of radius R when rotating at a rate of o about its axis in the outer-space, can simulate the same gravitational g, as living on earth if Ro2-g Show that, inside such a cylinder, when projectiles are shot in any arbitrary direction at low speeds (v<< Ro) and to low altitudes at nearby points (Ar<R), the equation of motion of such trajectories (as observed by a an...
(11 points) A uniform solid sphere of mass m and radius r is placed inside a hemispherical bowl of radius R. The sphere is released from rest at an angle theta and rolls without slipping. (a) (6 points) Using Conservation of Energy, to find an expression for the angular speed of the sphere when it reaches the lowest point of the bowl. (b) (6 points) Find the magnitude of the centripetal acceleration of the center of mass of the sphere...
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