A 45.7-kg boy on a swing moves in a circular arc of radius 3.80 m.At the lowest position, the child's speed reaches 2.78 m/s. Determine the magnitude of the tension in each of the two vertical support chains
If he sits in the middle of the swing, the tension force on each support chain will be equal. Call it T.
At the lowest position,
2T - weight = centripetal force
= M V^2/R.
2T = M*(g + V^2/R)
Solve for T.
M = 45.7 kg
R = 3.80 m
V = 2.78 m/s
g = 9.80 m/s^2
Well, let's begin with some swinging physics! It sounds like this boy is having a swinging good time.
Now, to determine the magnitude of the tension in each of the two vertical support chains, we'll need to consider the forces acting on the boy at the lowest position. At this point, the tension in the chains provides the centripetal force to keep the boy moving in a circular path.
The centripetal force (Fc) can be calculated using the formula Fc = mv²/r, where m is the mass of the boy, v is his speed, and r is the radius of the circular path.
Plugging in the given values, we have:
Fc = (45.7 kg)(2.78 m/s)² / 3.80 m
And after some arithmetic, we find that the centripetal force is approximately (wait for it...) 95.82 N. So, the total force exerted by both chains combined is about 95.82 N.
Since there are two support chains, we can divide this total force equally between them. Therefore, the magnitude of the tension in each vertical support chain is around... drumroll, please... 47.91 N.
Well, I hope that helped! Keep swinging and don't forget to enjoy the ride!
To determine the magnitude of the tension in each of the two vertical support chains, we need to consider the forces acting on the boy at the lowest position of the swing.
At the lowest position, there are two forces acting on the boy: the tension in each chain and the weight of the boy.
1. Weight: The weight of the boy is given by the formula: weight = mass * gravitational acceleration. In this case, the mass of the boy is 45.7 kg, and the acceleration due to gravity is approximately 9.8 m/s². So, the weight of the boy is:
weight = mass * gravitational acceleration = 45.7 kg * 9.8 m/s² = 448.36 N
2. Centripetal force: At the lowest position, the boy moves in a circular arc with a radius of 3.80 m. The centripetal force required to keep the boy moving in this circular path is given by the formula:
centripetal force = mass * velocity² / radius
In this case, the mass of the boy is 45.7 kg, and the velocity is 2.78 m/s (given). The radius is 3.80 m. So, the centripetal force is:
centripetal force = 45.7 kg * (2.78 m/s)² / 3.80 m = 96.29 N
According to Newton's second law, force = mass * acceleration. Therefore, the sum of the tension in the two vertical support chains and the weight of the boy must be equal to the centripetal force:
2 * tension + weight = centripetal force
Plugging in the values we calculated, we get:
2 * tension + 448.36 N = 96.29 N
Solving for tension:
2 * tension = 96.29 N - 448.36 N
2 * tension = -352.07 N
tension = -176.04 N
Since tension cannot be negative, it means there is an error in the calculation or the given information. Please recheck the values provided and confirm if there are any additional details that need to be considered.
To determine the magnitude of the tension in each of the two vertical support chains, we need to consider the forces acting on the boy at the lowest position of the swing.
At the lowest position, the boy's speed is 2.78 m/s. This speed can be considered as the centripetal speed, which is the speed required for the boy to stay in circular motion. The centripetal force (Fc) acting on the boy is provided by the tension in the support chains.
We can use the equation for centripetal force:
Fc = (mass * speed^2) / radius
First, we need to convert the mass of the boy from kg to N (Newtons) using the formula:
Weight = mass * acceleration due to gravity
Weight = 45.7 kg * 9.8 m/s^2 (assuming the acceleration due to gravity is 9.8 m/s^2)
Weight = 447.86 N
Now, we can calculate the centripetal force:
Fc = (mass * speed^2) / radius
Fc = (447.86 N * (2.78 m/s)^2) / 3.80 m
Fc = 925.16 N
Since there are two support chains, the total tension in the chains is twice the centripetal force:
Tension = 2 * Fc
Tension = 2 * 925.16 N
Tension ≈ 1850.32 N
Therefore, the magnitude of the tension in each of the two vertical support chains is approximately 1850.32 N.