4A Homework Set 4 (Newton's Laws with friction and/or circular motion)

1.  A horizontal force, F, pushes a block of mass m against a vertical wall. The coefficient of static friciton is given. Find the minimum horizontal force where the block will just begin to slip.

2. A ball of mass m is attached to a rigid vertical rod by means of two massless strings each a length L. Both strings are attached on the rod a vertical distance D apart. The rod, strings, and ball are rotated at some speed and both strings are taught. If the tension in the upper string is given as F_T, find the speed of the ball and the tension in the lower string.
 
3. A mass m on a frictionless table is attached to a hanging mass M by a cord through a hole in the table; the hole has no frictional effect on the string. Find the condition (the speed of the mass and the radius of its circular motion) with which it must spin for M to remain at rest.

4. A block of mass m at the end of a string is whirled around in a vertical circle of radius R. Find the critical speed below which the string would just become slack at the highest point".

5. A circular curve of highway is designed for traffic moving at a speed V. If the radius of the curve is r, what is the correct angle of banking of the road such that no friction force would be needed to negotiate it?

6. With the coefficient of static friction between the floor of a truck and a box resting on given, if the truck has an initial velocity of V_i , what is the least distance in which the truck can stop if the box is never to slide?

7. In an amusement park ride, riders stand against the vertical wall of a spinning cylinder. The floor falls away and the riders are held up by friction. If the radius of the cylinder is R, find the minmimum number of revolutions per time necessary if the coefficient of friction between a rider and the wall is given.

8.  Block M₁ is on top of block M₂ which is on top of a horizontal  frictionless surface.  A horizontal force, F, is applied to M₁.  The coefficient of static friction is u_s and the coefficient of kinetic friction is u_k. Find, a) the maximum force that can be applied to M₁, such that the position of M₁ relative to M₂ will not change.  (M₁ will not slip on top of M₂).  Then, if the force acting on M₁ is doubled, or F' = 2F, find the acceleration of each mass.

9.  A block of mass M₁ is resting on an incline of angle theta and is attached to a second block of mass m by a cord that passes over a smooth peg.  Mass m hangs vertically.  The coefficient of static friction between the block and the incline is u_s.  Find the range of possible values for m for which the system will be in static equilibrium.