Random Mechanics Problems

A rigid tube is placed in the vertical plane. It has the shape of an arbitrary curve, but has start and end points of the same height. A chain with uniform mass distribution lies in the tube. Show that the chain lies stationary in the tube, no matter its shape.

Solution can be found here.

A stick of mass density per unit length  rests on a circle of radius R. The stick makes an angle  with the horizontal and is tangent to the circle at its upper end. Friction exists at all points of contact, and assume that it is large enough to keep the system at rest. Find the friction force between the ground and the circle.

Solution can be found here.

A large number of sticks (with mass density per unit length p and circles (with radius R) lean on each other. Each stick makes an angle with the horizontal and is tangent to the next circle at its upper end. The sticks are hinged to the ground, and every other surface is frictionless. In the limit of a very large number of sticks and circles, what is the normal force between a stick and the circle it rests on, very far to the right? Assume that the last circle leans against a wall, to keep it from moving.

Solution can be found here.

Questions for Mechanics (1)

CONSEQUENCES OF NEWTON’S LAWS 

In reference to inertial reference frame X : Point mass A, 3kg, travels right at 5m/s. Point mass B,4kg, travels left at 20 m/s.

  1. What is the total momentum in the system in reference to frame X?
  2. A frame Y is defined as moving leftwards at velocity V relative to frame X. In frame Y, the sum momentum of masses A and B is zero. What is the value of V, and what is the momentum of mass B in reference to frame Y?

A man of mass 100kg stands at the leftmost tip of a stationary boat with mass 700kg. The boat is 16m in length. If the man walks to a point 2m from the right end of the boat, what is the total displacement of the boat?

FORCES AND FREE-BODY DIAGRAMS

A ladder is left stationary on a rough floor, leaning on a rough wall. It has a mass M.

  1. What is the coupled action-reaction pair force corresponding to the weight of the ladder?
  2. Draw a free body diagram of the ladder, displaying all forces at both points of contact of the ladder with the floor and wall. [hint : include the weight of the ladder, and all frictional and normal forces]
  3. The coefficient of friction of the ladder against the floor and wall are μf and μw, respectively. Let the normal forces at the floor and wall be Nf and Nw, respectively. Assuming that the ladder is just at the equilibrium point before slipping at both contact points, form two equations relating :
    1. all the horizontal forces involved.
    2. all the vertical forces involved.
  4. If μf = 0.6 and μw = 0.4, What are the values of Nw and Fw?

A ladder is left stationary on a rough floor, leaning against a perfectly smooth wall. It has an even mass distribution along its length L, and a mass of M. It forms an angle θ with the horizontal. Let its point of contact with the ground be G.

  1. Draw a free body diagram of the ladder, displaying all forces at both points of contact of the ladder with the floor and wall.
  2. Express the turning moment of the ladder about G, due to Nw.
  3. Express the turning moment of the ladder about G, due to the weight of the ladder.
  4. Noting that the system is in rotationary equilibrium, express Nw in terms of θ and its weight.
  5. Let the mass of the ladder be 5kg. If θ = 60 °, what is the value of Nw?

 

 

A Random Mechanics Question

(adapted from STEP I, 2009 Q9)

A child plays with a toy cannon on the floor of a long railway carriage, moving north with the acceleration a. The cannon is pointed south at an angle θ to the horizontal. It fires a shell at speed v.

  1. Give an expression for the range of the shell in the carriage. [in terms of v, θ, g and a]
  2. Find the value of tan(2θ) that maximises the range of the shell in the carriage. [hint : its just differentiation]