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S2.7 Tension

[A] About the force

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Tension does not really have the credentials of a force: it has a ‘magnitude’ (of the right ‘units’!), but not a direction. It is best to think of it as something which can be used to specify a force–namely the force which a stretched object exerts on the point to which it is attached. The example that follows may help.
 

Commentary

Commentary

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About the ‘F-symbols’

About the ‘F-symbols’

We could think of all the forces that feature in this argument as ‘one-dimensional vectors’ whose sign prescribes their direction (see S0.4). But here it is clearer if we use the ‘F-symbols’ to represent only the magnitudes of the forces and put in the appropriate signs (plus or minus) explicitly, when we come to combine the forces.
 
  • Apply Newton’s 2nd law to the string:
    FSB-FSA=ma
  • The RHS is zero if string is light.
  • Then FSA=FSB
  • Invoke the 3rd law (twice):
    \[ F_{SA}=F_{AS} \hspace{1cm}\mbox{\rm and}\hspace{1cm} F_{SB}=F_{BS} \]
  • So string exerts forces of equal magnitude on each end:
    FAS=FBST
 

[B] Example problem

Two masses m1 and m2 are suspended over a light and frictionless pulley. Find the accelerations of the masses and the string tension.
Graphic - No title - attwood

Solution

Solution

  • Consider mass m1
    • Let a be downward acceleration
    • Newton’s 2nd law (applied to m1):
      m1g-T=m1a
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  • Consider mass m2
    • Its upward acceleration is a
    • Newton’s 2nd law (applied to m2):
      T-m2g=m2a
Graphic - No title - attwoodtwo
  • Add equations:

    (m1-m2)g=(m1+m2)a

    so

    $$a= \frac{m_1- m_2}{m_1 +m_2} g$$

  • Substitute:

    \[ T= m_2 (a+g) =\xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{m_2 g \left[ \frac{m_1-m_2}{m_1 +m_2} +1 \right]} \]

    so

    $$T=\frac{2m_1 m_2}{m_1 +m_2} g$$

 

Results

string tension:

\[ T= \frac{2m_1 m_2}{m_1 +m_2} g \]
downward acceleration of m1 (and upward acceleration of m2): 
\[ a= \frac{m_1- m_2}{m_1 +m_2} g \]

Check It!
TSpring Tension

Learning Resources

Textbook: HRW Chapter 5.7,5.9
Course Questions:
Self-Test Questions: