# Why are the two acceleration considered as a1 and a2 here and find acceleration of 6kg mass?

Why are the two acceleration considered as a1 and a2 here? What is the reason for this even though the tension of both the strings are same. T is equal to 5gTcos 53 for left side and 5gTcos37 for the right side

• How much lifting work is being done on m1 and m2 to raise their potential energy? Consider the extreme cases where plane 1 was horizontal and plane 2 was vertical? (I presume the sliding surfaces are frictionless.) – Transistor Jan 23 at 14:21
• @Transistor the sliding surfaces are frictionless. In your extreme case , the mass m1 and m2 will definitely move. – srijan Sri Jan 23 at 14:22
• @Transistor is it because the components of gravity will be different right – srijan Sri Jan 23 at 14:23
• @Transistor any idea on how can we find acceleration for 6kg mass – srijan Sri Jan 23 at 14:24
• I haven't done this stuff for many, many years. Start with your initial conditions. Edit your question to show what the initial value of T is. – Transistor Jan 23 at 14:25

As posed, you have three unknowns - $$a1$$, $$a2$$, and $$a3$$. The tension on the cable is everywhere uniform, but the accelerations are different because the static force components affecting the tension are different.

$$F=ma$$ at each mass. So for mass $$m1$$, the tension is
$$m1*(g*cos53 +a1)$$. Likewise
$$m2*(g*cos37 + a2)$$ and
$$1/2*m3*(g*cos90 +a3)$$.

These are all equal. And the length of the line doesn't change.

You can assume that you start from a stationary condition (all velocities are initially zero) - that doesn't change the accelerations. You can choose any initial length that like - that doesn't change the accelerations either.

So $$m1(g*cos53 + a1) = 1/2 m3(g*cos90 + a3) = m2(g*cos37 +a2)$$ from equal tension.

And $$a1 = 2*a3 - a2$$ from constant length.

$$5 kg (0.600g + a1) = 3 kg (1g + a3) = 5 kg ( 0.799g + a2)$$
$$0.01 kg + 5 kg * a1/g = 3 kg * a3/g = 0.993 kg + 5 kg * a2/g$$
$$0.0033 + 1.666 a1/g = a3/g = 0.331 + 1.666 a2/g$$

$$0.0033 + 1.666 (2a3 - a2)/g = a3/g$$
$$0.0033 - 1.666 a2/g = -2.332 a3/g$$
$$-0.0014 + 0.7144 a2/g = a3/g$$

$$-0.0014 + .7144 a2/g = 0.331 + 1.666 a2/g$$
$$-0.3324 = .9516 a2/g$$

$$a2 = -0.349 g$$
$$a3 = -0.2509 g$$
$$a1 = -0.1529 g$$