APPC Unit 2
Challenge Problems 1: Equilibrium
1. A block of mass m1 on a frictionless inclined plane of angle α is
connected by a cord over a massless, frictionless pulley to a
second block of mass m2 hanging vertically.
a) Draw a well-labeled free-body diagram showing all the
forces acting on each object.
b) Derive a relationship between m1, m2, and α that would allow
the system to sit in equilibrium
c) Friction is now considered and the coefficient of static friction is µs. Draw a new FBD to
represent the following two cases. If the system remains at rest, then in terms of m1, µs, α and
determine
i. the minimum hanging mass, mmin, and
ii. the maximum hanging mass, mmax.
d) The set-up is now changed such that the same block 1 slides at constant speed up the same
angled incline. If block 2 has a mass of mup, then derive an expression for the coefficient of
kinetic friction µK.
e) If block 1 is to slide down the incline at a constant speed, then find an expression for the
maximum value of the coefficient of kinetic friction for which this is possible if there is no
hanging block.
2. A horizontal force F pushes a block of mass M against a vertical wall. The coefficient of static
friction between the wall and the block is µS, and the coefficient of kinetic friction is µK. Assume that
the block does not move.
a) Draw a well-labeled free-body diagram showing all of the forces
acting on the block.
b) Derive a relationship for the minimum pushing force in terms of M
and µS that would allow the block to remain in static equilibrium.
c) The pushing force is now angled up (but still into the wall) at an
angle θ relative to the horizontal. Derive an expression for the
minimum magnitude of the pushing force that maintains static equilibrium in terms of M, µS,
and θ.
d) The pushing force is now angled below horizontal at an angle θ.
i. Derive a similar expression as in part (c) but assume the block moves at a constant
speed down the wall.
ii. Is there a minimum value for the angle at which no amount of force would move the
block at constant speed? If so, what is it, if not, explain why not?
This study source was downloaded by 100000899606070 from CourseHero.com on 06-19-2025 09:49:54 GMT -05:00
https://www.coursehero.com/file/215772923/16-APPCM-Challenge-Problems-1pdf/
Challenge Problems 1: Equilibrium
1. A block of mass m1 on a frictionless inclined plane of angle α is
connected by a cord over a massless, frictionless pulley to a
second block of mass m2 hanging vertically.
a) Draw a well-labeled free-body diagram showing all the
forces acting on each object.
b) Derive a relationship between m1, m2, and α that would allow
the system to sit in equilibrium
c) Friction is now considered and the coefficient of static friction is µs. Draw a new FBD to
represent the following two cases. If the system remains at rest, then in terms of m1, µs, α and
determine
i. the minimum hanging mass, mmin, and
ii. the maximum hanging mass, mmax.
d) The set-up is now changed such that the same block 1 slides at constant speed up the same
angled incline. If block 2 has a mass of mup, then derive an expression for the coefficient of
kinetic friction µK.
e) If block 1 is to slide down the incline at a constant speed, then find an expression for the
maximum value of the coefficient of kinetic friction for which this is possible if there is no
hanging block.
2. A horizontal force F pushes a block of mass M against a vertical wall. The coefficient of static
friction between the wall and the block is µS, and the coefficient of kinetic friction is µK. Assume that
the block does not move.
a) Draw a well-labeled free-body diagram showing all of the forces
acting on the block.
b) Derive a relationship for the minimum pushing force in terms of M
and µS that would allow the block to remain in static equilibrium.
c) The pushing force is now angled up (but still into the wall) at an
angle θ relative to the horizontal. Derive an expression for the
minimum magnitude of the pushing force that maintains static equilibrium in terms of M, µS,
and θ.
d) The pushing force is now angled below horizontal at an angle θ.
i. Derive a similar expression as in part (c) but assume the block moves at a constant
speed down the wall.
ii. Is there a minimum value for the angle at which no amount of force would move the
block at constant speed? If so, what is it, if not, explain why not?
This study source was downloaded by 100000899606070 from CourseHero.com on 06-19-2025 09:49:54 GMT -05:00
https://www.coursehero.com/file/215772923/16-APPCM-Challenge-Problems-1pdf/
