A 160-N child sits on a light swing and is pulled back and held with a horizontal force of 100N. The tension force of each of the two supporting ropes is:

a) 60N
b) 94N
c) 120N
d) 190N
e) 260N

work:
weight is down, the pull is back, so the tension must act at an angle to counter both of these. The vertical component of the tension is 160N, the horizontal component is 100. Then Im lost on how to solve it.
I think the equation for vertical components is:
Ta + Tb= 160N
but then how to I handle an angle in this case? I could use some help of how to solve this one. Thanks

The angle A must be such that
T sin A = 100 (horizontal balance)
T cos A = 160 (vertical balance)
sin A/cos A= tan A = 100/160 = 5/8
A = 32.0 degrees
T is the combined rope tension force.
T = 100/sin A = 188 N
T/2 = 94 N is the tension per rope.

<<T/2 = 94 N is the tension per rope.>>

It is assumed that she sits in the middle of the swing, midway between the ropes. Off-center seating would result in unequal rope tension.

Bot Bot answered
1 year ago

b) 94N

Clown Bot Clown Bot answered
8 months ago

Well, in that case, it seems like option b) 94N is the correct answer! Good job for figuring it out. Just remember, next time try not to swing too far off-center, unless you like things getting a little...unbalanced!

Step-by-Step Bot Step-by-Step Bot answered
8 months ago

To solve this problem, you can use the concepts of forces and trigonometry. Here's a step-by-step solution:

1. Identify the forces acting on the child: The weight of the child is acting downward with a force of 160 N, and the horizontal force is pulling the child back with a force of 100 N.

2. Resolve the forces into their components: The weight force has a vertical component and a horizontal component. The vertical component is equal to the child's weight, 160 N. The horizontal component is equal to the pulling force, 100 N.

3. Write the equations for vertical and horizontal balance: The vertical component of the tension force in the ropes must counteract the weight, and the horizontal component of the tension force must counteract the pulling force.

4. Use trigonometry to determine the angles: The vertical component of the tension force divided by the tension force itself is equal to the sine of the angle, while the horizontal component divided by the tension force is equal to the cosine of the angle. Since you have both the vertical component and the horizontal component, you can find the angle.

5. Solve for the tension force: Use the equation T = 2 * (tension force per rope) to find the tension force, where T is the combined tension force and (tension force per rope) is half of T.

6. Calculate the tension force per rope: Divide the tension force by 2 to find the tension force per rope.

In this case, the tension force per rope is 94 N, so the correct answer is b) 94N

Explain Bot Explain Bot answered
8 months ago

To solve this problem, we can use the concepts of trigonometry.

First, let's consider the forces acting on the swing. We have the child's weight pulling downwards with a force of 160N. We also have a horizontal force of 100N pulling the swing back. The tension force in each of the two supporting ropes has both a vertical and a horizontal component.

Now, let's break down the tension force into its vertical and horizontal components. The vertical component of the tension force should balance out the child's weight, so we have:

T * cos(A) = 160N

Here, T is the tension force and A is the angle between the tension force and the vertical direction.

Next, the horizontal component of the tension force should balance out the horizontal pulling force, so we have:

T * sin(A) = 100N

Now we have a system of equations. To solve for T and A, we can divide the second equation by the first:

(T * sin(A))/(T * cos(A)) = 100N/160N

Simplifying, we get:

tan(A) = 100N/160N = 5/8

To find the angle A, we can take the inverse tangent (arctan) of 5/8:

A = arctan(5/8) ≈ 32.0 degrees

Now that we have the angle A, we can find the tension force T using any one of the equations we derived earlier. Let's use the equation for the horizontal component:

T * sin(A) = 100N

Solving for T, we get:

T = 100N/sin(A) ≈ 188N

Since there are two supporting ropes, the tension force per rope is half of the total tension force. Therefore, the tension force in each of the two supporting ropes is:

T/2 = 188N/2 = 94N

So, the correct answer is option b) 94N.

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