A 650 N boy and a 490 N girl sit on a 150 N porch swing that is 1.70 m long. If the swing is supported by a chain at each end, what is the tension in each chain when the boy sits .75 m from one end and the girl .5 m from the other end?
So then assuming the swing is uniform, do I have:
T1 + T2 = 650 + 490 + 150 = 1290 N
Then,
T1*1.7 -0.95*650 - 0.5*490 -.85*150 = 0
Solve for T1 and then use the first equation to solve for T2?
Yes. :-)
To find the tension in each chain, we need to apply the concept of torques. Torque is the rotational equivalent of force and can be calculated as the product of force and the perpendicular distance from the point of rotation.
In this case, the porch swing is supported by two chains at each end, creating a rotational situation. The sum of the individual torques acting on the swing should be zero for it to be in rotational equilibrium.
Let's consider the boy first. The torque due to the boy's weight can be calculated by multiplying his weight (650 N) by the perpendicular distance between his position and the point of rotation (0.75 m). Similarly, the torque due to the girl can be obtained by multiplying her weight (490 N) by her perpendicular distance from the other end (1.70 m - 0.5 m = 1.20 m).
To find the tension in the chain, we divide the sum of the torques by the length of the swing (1.70 m). Since the swing is in equilibrium, the sum of the torques acting on it must be zero. Thus, the torque due to the boy's weight must be equal and opposite to the torque due to the girl's weight.
Let's calculate the torques first:
Torque due to the boy: Torque_boy = 650 N * 0.75 m = 487.5 Nm
Torque due to the girl: Torque_girl = 490 N * 1.20 m = 588 Nm
Since the swing is in equilibrium, the sum of the torques is zero:
Torque_boy + Torque_girl = 0
487.5 Nm + 588 Nm = 0
Now, we can solve for the tension in each chain:
Tension_boy * 1.70 m - Tension_girl * 1.70 m = 0
We know that the tension in each chain is the same, so we can represent it as Tension:
Tension * 1.70 m - Tension * 1.70 m = 0
Tension * (1.70 m - 1.70 m) = 0
Tension * 0 m = 0
The equation simplifies to Tension * 0 = 0, which is true for any value of tension. Hence, we cannot determine the specific tension in each chain based on the given information.
In this scenario, since the sum of the torques due to both individuals' weights cancels out, the tension in each chain is indeterminate. The problem requires additional information to calculate the exact tension in each chain.
The sum of the two chain tesnions is
T1 + T2 = 650 + 490 = 1140 N
Take moments about the chain nearest to the girl and set the sum equal to zero. Call the cable nearest the boy "1". The boy is 0.95 m from the girl's end, "2".
T1*1.7 -0.95*650 - 0.5*490 = 0
Solve for T1. Then use T1 + T2 = 1140 N to solve for T2.