Interacting objects, Newton's 2nd and 3rd laws, conservation laws

Newton's second law

Problem:

Two masses (m1 = 15 kg and m2 = 20 kg) are connected by a massless cord and placed on a horizontal, frictionless surface.  The two-mass system is pulled to the right by a force FA = 60 N using a cord that makes an angle of 40 degrees with the horizontal.  The masses remain on the horizontal surface.  There is no vertical motion.
image
(a)  Draw the free body diagrams for the two masses, properly labeling all forces.
(b)  What is the normal force between mass m2 and the horizontal surface?
(c)  What is the acceleration of the system?
(d)  What is the tension FT in the cord between the two masses?

Problem:

A 100 kg man climbs a vertical rope with an acceleration of 12 cm/sec2.  Find the tension in the rope.

Solution:

Problem:

The force F in the figure below pushes a block of mass M = 5 kg, which in turn pushes a block of mass m = 1 kg.  There is no friction between M and the supporting surface.  If the friction coefficient between the two blocks is μ, how large must F be if the block of mass m is not to slip?

 

image

 

Solution:

Problem:

Consider the system of 3 masses shown below.

image

Assume all surfaces to be frictionless and the inertia of pulley and cord to be negligible.  Find the horizontal force necessary to prevent any relative motion of M, m1, and m2.

Solution:

Problem:

Near the surface of Earth, two masses, m1 = 1 kg and m2 = 3 kg, are connected by a string of negligible mass.  The masses hang on opposite sides of a pulley with radius R.  The pulley is fixed, and the string can slide over the pulley.
(a)  Find the acceleration of the masses if there is no friction between the string and the pulley.
(b)  Find the acceleration of the masses if the coefficient of kinetic friction between the string and the pulley is μ = 0.1.

image
Solution

Problem:

Consider a glass that is resting on top of a tablecloth.  The glass has mass m1 and the tablecloth has mass m2.  Suppose the coefficient of static friction between the tablecloth and glass is given by μs, and the table is highly polished so it is nearly frictionless.
What is the maximum force that the tablecloth can be pulled horizontally so that the glass and table cloth move together without slipping?

Solution:

Problem:

A light flat ribbon is placed over the top of a triangular prism as shown in the diagram.   Two blocks are placed on the ribbon.   The coefficients of static and kinetic friction between the ribbon and the blocks are μs and μk, respectively.  There is no friction between the ribbon and the prism.  The angle θ and the masses of the blocks m and M are given.  Assuming that M > m, find the acceleration of the ribbon along the prism after the blocks are simultaneously released.  Consider all possible cases.

image

Solution:


Newton's third law

Problem:

Sandra, who has a mass m = 40 kg stands on a M = 28 kg flatboat.  Her distance to the shore is 9.4 m.  She walks 2.6 m along the boat toward the shore and then stops.  How far is she away from the shore?  Assume there is no friction between the boat and the water.

Solution:

Problem:

A cat and a dog are sitting on the top of a small wagon which is a rest.  The mass of the dog is 20 kg and the mass of the cat is 5 kg.  There is no friction between the wagon and the ground.  The cat jumps from the wagon and then the dog jumps in the same direction from the already moving wagon.  When the dog jumps off, the wagon's velocity increases seven fold.  The horizontal components of the dog's and the cat's velocities before touching the ground are the same.  What is the wagon mass?

Solution:


Conservation laws

Problem

Two blocks of mass M and 3M are placed on a horizontal frictionless surface.  A light spring is attached to one of them, and the blocks are pushed together with the spring between them.  A cord holding them together is burned, after which the block of mass 3M moves to the right with a speed of 2 m/s.
(a) What is the speed of the block of mass M?
(b) Find the original elastic potential energy in the spring if M = 0.35 kg.

image

Solution:

Problem:

If two objects collide and one is initially at rest, is it possible for both to be at rest after the collision?  Is it possible for one to be at rest after the collision?  Explain!

Solution:

Problem:

A sled on which you are riding is given an initial push and slides across frictionless ice.  Snow is falling vertically (in the frame of the ice) on the sled.  Assume that the sled travels in tracks which constrain it to move in a straight line.  Which of the following three strategies causes the sled to move the fastest?
(i)  You sweep the snow off the sled so that it leaves the sled in the direction perpendicular to the sled's tracks, as seen by you in the frame of the sled.
(ii)  You sweep the snow off the sled so that it leaves the sled in the direction perpendicular to the sled's tracks, as seen by someone in the frame of the ice.
(iii) You do nothing.
Order the strategies from best to worst and explain the ordering.

Solution:


Problem:

A particle of mass m strikes a frictionless, smooth, hard surface at an angle θ from the surface normal.  The particle bounces off with coefficient of restitution ε.  (The coefficient of restitution is defined as the ratio of the magnitudes of the velocity components along the normal to the plane of contact after and before the collision.)  Find the rebound angle for the particle as it leaves the surface.

Solution: