Circular motion, centripetal acceleration, in a uniform gravitational field

Problem:

A small mass slides without friction down the loop track shown in the figure below.
(a)  Show that the speed at point B must be at least as large as (gR)½ if the mass does not fall away from the track.
(b) What must be the height h required to achieve the speed found in (a)?  Give your answers in terms of R.

 

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Solution:

Problem:

A ball is weighing 4 pounds is suspended by a weightless string of length 6 ft.  The ball is delivered a horizontal impulse of magnitude 3 pounds-seconds.  Find the maximum height above its initial position, attained by the ball.  Take 32 ft/s2 as the gravitational acceleration and assume there is no friction at the point of attachment of the string.

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Solution:

Problem

A child slides down a frictionless slide as shown in figure.
(a) What is the minimum value of R for the child to not immediately loose contact with the section of radius R?
(b) If R is larger than that minimum value at what height h will the child loose contact with the section of radius R?

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(c)  If R is smaller than that minimum value, how far from the end of the section with radius R will the child land?

Solution:

Problem:

A massless string of length L has a ball of mass m attached to one end and the other end is fixed.  The ball is launched so that it moves in a vertical circle in a gravitational field, with an initial velocity v0 downward.  What is the minimum value for v0 if the ball is to rotate around on a circle of radius L?

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Solution:

Problem:

A particle attached to the end of a light (massless) string of length r suspended from a very thin rod.  Initially the particle is at rest.  It is given an horizontal impulse which results in an horizontal velocity v0 right after the impulse.  What is the minimum speed v0 that allows the string to wrap around the pivot rod at least once?

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Solution:

Problem:

A block of mass m starts from rest at θ = 0 and slides without friction under the force of gravity on an inverted hemispherical surface of radius R.  Find the angle θ when the block leaves the surface.

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Solution:

Problem

A wooden block of mass M hangs on a massless rope of length L.  A bullet of mass m collides with the block and remains inside the block.  Find the minimum velocity of the bullet so that the block completes a full circle about the point of suspension.

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Problem:

A car of total mass M1 = M and velocity v1 makes a totally inelastic collision at time t = 0 with a second car of mass M2 = 2M at rest.  Before the collision a point object of mass m << M was sitting at the bottom of a frictionless spherical cavity of radius r embodied inside the first car.  For what range of velocities v1 will the small mass lose contact with the surface of the cavity