Connected Bodies - Ropes & Tow Bars (CIE A Level Maths: Mechanics)

What are connected bodies/particles?

• The phrase connected particles refers to situations where two (or more) bodies (objects) are connected in some way.
• Common examples include:
  • a car towing a caravan or trailer
  • a load being raised by a lift (3.2.3)
  • two bodies connected by a rope that passes over a pulley (3.2.4)

• Problems may involve the particles being stationary (in equilibrium) or in motion – in the latter case Newton’s Laws of Motion will be involved.

What are Newton’s Laws of Motion?

• Full details of Newton’s Laws of Motion can be found in 3.2.1 F = ma but Newton’s Third Law of Motion (N3L) is particularly relevant for the problems covered in this note
  • For two bodies, the force exerted on the second body by the first is equal in magnitude but opposite in direction to the force exerted on the first body by the second

What are ropes and how are they modelled?

• A rope is typically used to connect two inanimate objects such as blocks, crates, containers, etc
• A rope would be modelled as a light inextensible string
  • The modelling assumption light means the rope’s mass is so small relative that it can be ignored
    • Mathematically this means that the tension in the string is constant throughout its length (i.e. tension is equal on both sides of the string)

  • The modelling assumption inextensible means the rope cannot be extended/shortened in length
    • Mathematically this means that both connected particles will have the same acceleration

  • A string would only be in tension (not thrust – see tow bars for thrust)
  • A string can go slack – for example if one particle is disconnected – in which case the model being used would no longer apply and a new scenario would ensue with no tension involved

  

What are tow bars and how are they modelled?

• A tow bar is a mechanism by which a car (or similar vehicle) can be connected to a caravan, trailer (or similar)
• A tow bar is modelled as a light (inextensible) rod
  • A rod can either be in tension or thrust (compression)
    • For a car towing a caravan by a light rod, the rod would be in tension when the car is accelerating, thrust when it is decelerating

    

What is a coupling?

• A coupling is a general term referring to the connection between two objects - usually a relatively complex system, such as how two train carriages are connected - but for modelling purposes is simplified to a string or rod

How do I solve problems involving tow bars and ropes?

• If a particle is in motion in the direction being considered, then Newton’s Laws of Motion apply so use “F = ma” (N2L)
• If a particle is not in motion in the direction being considered then “F = 0” can be used, although
• “F = ma” with “a = 0” will also work
• Step 1.Draw a series of diagrams,
  • Label the forces and the positive direction of motion.
  • Colour coding forces acting on each particle may help.

• Step 2. Write equations of motion, using “F = ma ” (or if no motion “F = 0”)
• Step 3. Solve the relevant equation(s) and answer the question
  • • Some trickier problems may lead to simultaneous equations

• If both particles are travelling in the same direction the system can be treated as one particle (as well as separate particles)
  • • There is no tension at either side of the string when the system is treated as one - mathematically they cancel each other out

• For constant acceleration the ‘suvat’ equations could be involved

 a m s^-2 is the acceleration of the system

 m₁ kg and m₂ kg  are the masses of the two bodies

 m₁ g N and m₂ g N are the weights of the two bodies

 T N is the tension in the string

 D N is the driving force of the system

 F₁ N and F₂ N are the resistive forces acting on the two bodies

 R₁ N and R₂ N  are the normal reaction forces of the two bodies

* You do not necessarily need all diagrams but if in doubt draw all as they may help you to understand the problem more clearly **

How do we form the equations for problems involving tow bars and ropes?

• Form the equations as follows:
  • Treating the particles as one

 Horizontally (→) D - (F₁ + F₂) = (m₁ + m₂)a

                               There is no vertical motion so use “F = 0”

            (↑) (R₁ + R₂) - (m₁ + m₂)g = 0

   (F= ma with a =0  will lead to the same equation)

• • Treating each particle separately

Particle 1:            Horizontally (→) T - F₁ = m₁a                                                                     
                              Vertically(↑) R₁ -m₁g = 0 (No motion)

Particle 2:            Horizontally (→) D - T- F₂ = m₂a                                                             
                              Vertically(↑) R₂ - m₂g = 0      (No motion)

• You do not necessarily need all equations but if in doubt attempt all and it may help you make progress

(a)  Find the engine force from the plane.

(b)  Find the tension in the tow rope.