Equilibrium

Problem:

A 1500 kg automobile has a wheelbase (the distance between the axles) of 3 m.  The center of mass of the automobile is on the centerline at a point 1.2 m behind the front axle.  Find the force exerted by the ground on each wheel.

Solution:

Problem:

A board weighing 90 N and 12 m long rests on two supports with each support being one meter from its respective ends.  A 360 N block is placed on the board 2 m from the rightmost support. Find the force that is exerted by each support on the board.

Solution:

Problem:

A uniform massive rod AB of length L rests at the inclination ±10o with respect to the horizontal with end A in contact with a vertical wall.  The two configurations are shown in figures a and b.  End of the rod is supported by a massless cord BC, which is of length 1.2 L and is attached at C, a point on the wall above A.  What coefficient of static friction between A and the wall is necessary to achieve each configuration?

image image

Solution:

Problem:

A homogeneous ladder of length L leans against a wall (see figure).  The coefficient of friction with the ground is μ1, the coefficient of friction with the wall is µ2.  Find the minimal angle θ, below which the ladder will slip.

image

Solution:

Problem:

A long thin uniform rod lies flat on the table as shown.  One end of the rod is slowly (quasi-statically) pulled up by a force that remains perpendicular to the rod at all times.  What minimum coefficient of static friction is required so that the rod can be brought to the vertical position without any slipping of the bottom end?

image

Solution:

Problem:

Two steel balls, each of radius R and mass M, are packed into a bottomless cylindrical thin-walled tube of diameter R(2 + √2).  The tube and balls sit on a table.  There is no friction anywhere in the problem.  Find the minimum mass of the tube m that will keep the system in equilibrium.
image

Solution:

Problem:

A uniform metal plate has mass m and length 4R.  The plate is bent in half at right angles and placed on a horizontal cylinder of radius R as shown.  
image
The top half of the plate is horizontal.  The cylinder is fixed.  What is the minimum coefficient of static friction μs between the plate and the cylinder that allows the plate to remain at rest?

Solution:

Problem:

A rope is wound around a cylinder of radius R with a contact angle φ (φ can be larger than 2π) to hold a massive object in place, for example to secure a ship for docking.  The object exerts a force F on one end of the rope and the other end of the rope held by a force T.   The coefficient of static friction between the rope and the cylinder is k.  Find the ratio F/T required to keep the object stationary as a function of φ.

image

Solution:

Problem:

A rope rests on two platforms which are both inclined at angle θ (which you are free to pick) as shown.  The rope has uniform density and its coefficient of friction with the platform is 1.  The system has left-right symmetry.  What is the largest possible fraction of the rope that does not touch the platform?  What angle θ allows this maximum value?

image

Solution: