Tension in a rope can not be negative

Question

If !!m_1 < m_2!! and the floor is smooth. If the car accelerates towards right:

  • Rope will be under tension
  • Rope will be slack

Solution

There is friction between !!m_1!! and !!m_2!! which helps in accelerating.

We can solve it using 2 methods.

First suppose the rope is under tension. Let !!T!! be the tension in the string, !!F!! be the friction between !!m_1!! and !!m_2!!

So on !!m_1!!:

!!m_1 a=T+F!!

On !!m_2!!

!!m_2 a=F-T!!

So !!T= (m_1-m_2)a!!

But !!m_1 < m_2!! so !!T<0!! which is not possible since it is rope.

Method 2:

Since the same frictional force is acting on !!m_1,m_2!!. But !!m_1 < m_2!! and the same tension is acting on !!m_1,m_2!!. So the acceleration of !!m_1 > m_2!!, hence the rope will be slack and there will be no tension in the rope.

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