# Is tension equal to the weight of another object?

I have been solving this problem:

A 5kg object on the table is linked to a 3kg object hanging from a massless rope in a pulley system as shown in the picture. Find the acceleration of a 5-kg objected. Ignore friction

So I drew up my free-body diagram and came up with these equations:

I assumed tension is the same on

$$m_1a = T$$ $$m_2a = W-T$$

After I got those two equations I replaced T this equation:

$$m_2a = m_2g - m_1a$$

Then I did some algebra to simply the equation, plugged in the masses and solved for acceleration:

$$a = \frac{m_2g}{m_2+m_1}$$

Acceleration equaled to around 3.68m/s^2

However, my friend was arguing that my calculations were incorrect because he said that tension equals to the weight of object 2, therefore he said $$T = W_2$$ $$T = m_2g$$ $$T = 29.4N$$

After he obtained this tension he then solved for acceleration and got an acceleration of object 1 to be 5.88m/s^2

Is he correct to evaluate tension as the weight of object 2?

• If he was right why would the hanging mass accelerate? – M. Enns Dec 8 '16 at 2:49

First of all common sense tells us that if the surface is smooth enough, then the system is definitely going move with certain acceleration.

If so, then mass 2 will also accelerate, this implies that net force on m2 is not zero. But if you put T = W for mass 2, then net force will become zero. Thus the assumption that T=W for m2 is wrong.

And yes your assumption that m2a = W - T and m1a = T, are correct.

• Are my assumptions of m1a=T m2a=W−T correct? – Pablo Dec 8 '16 at 10:49
• Yes your assumption is absolutely correct. – Nisarg joshi Dec 8 '16 at 12:44

Please note your solution has implicitly used the assumption that the stiffness of the string is infinite so that the accelerations of the two bodies are the same.

Your friend is wrong and you are right. Since the system is accelerating the tension in the string must be less than the weight of the second object, as indicated by your second equation.

your assumption is correct since differecce between weight of the suspended mass and this rope tension gives the force needed to accelerate the system.