 340-N child is in a swing that is attached to a pair of ropes 1.90 m long. Find the gravitational potential energy of the child–Earth system relative to the child's lowest position at the following times.

(a) when the ropes are horizontal
 J

(b) when the ropes make a 34.0° angle with the vertical
 J

(c) when the child is at the bottom of the circular arc
 J




To find the gravitational potential energy (\( U \)) of the child–Earth system at various positions, we use the formula:

\[
U = mgh
\]

where:
- \( m \) is the mass of the child,
- \( g \) is the gravitational acceleration (approx. \( 9.8 \, \text{m/s}^2 \)),
- \( h \) is the height of the child relative to the lowest point of the swing.

Given dataTo find the gravitational potential energy (\( U \)) of the child–Earth system at various positions, we use the formula:

\[
U = mgh
\]

where:
- \( m \) is the mass of the child,
- \( g \) is the gravitational acceleration (approx. \( 9.8 \, \text{m/s}^2 \)),
- \( h \) is the height of the child relative to the lowest point of the swing.

Given 
- Weight (\( W = mg \)) of the child is \( 340 \, \text{N} \), so we can find \( m = \frac{340}{9.8} \approx 34.69 \, \text{kg} \).
- Rope length \( L = 1.90 \, \text{m} \).

We'll calculate the height \( h \) relative to the lowest point for each position and use it to find the potential energy.

### Part (a): When the ropes are horizontal

1. If the ropes are horizontal, the child is at the maximum height, equal to the length of the rope (1.90 m) above the lowest point.
   
2. So, \( h = 1.90 \, \text{m} \).

3. The gravitational potential energy at this point:
   \[
   U = mgh = (34.69 \, \text{kg})(9.8 \, \text{m/s}^2)(1.90 \, \text{m})
   \]

   Calculating this:

   \[
   U \approx 647.68 \, \text{J}
   \]

### Part (b): When the ropes make a 34.0° angle with the vertical

1. At an angle of \( 34.0^\circ \), the height \( h \) is given by:
   \[
   h = L - L \cos \theta = L(1 - \cos \theta)
   \]
   where \( \theta = 34.0^\circ \).

2. Plugging in the values:
   \[
   h = 1.90 \times (1 - \cos 34.0^\circ)
   \]

3. Using \(\cos 34.0^\circ \approx 0.829\):
   \[
   h \approx 1.90 \times (1 - 0.829) \approx 1.90 \times 0.171 \approx 0.325 \, \text{m}
   \]

4. Now, the gravitational potential energy:
   \[
   U = mgh = (34.69)(9.8)(0.325)
   \]

   Calculating this:

   \[
   U \approx 110.38 \, \text{J}
   \]

### Part (c): When the child is at the bottom of the circular arc

1. At the lowest position, \( h = 0 \) because the child is at the reference point.

2. Therefore, the gravitational potential energy:
   \[
   U = mgh = (34.69)(9.8)(0) = 0 \, \text{J}
   \]

### Summary of Answers:

- **(a)** \( U \approx 647.68 \, \text{J} \)
- **(b)** \( U \approx 110.38 \, \text{J} \)
- **(c)** \( U = 0 \, \text{J} \)

Would you like further explanation or clarification on any of these parts?

---

1. How does the angle affect the gravitational potential energy in such scenarios?
2. Why is potential energy zero at the lowest position of the swing?
3. What are some other ways to calculate gravitational potential energy in pendulum systems?
4. How would these values change if the swing rope length increased?
5. Why is cosine used to determine height in part (b)?

**Tip:** Gravitational potential energy depends only on the height relative to a reference point, so it's key to choose this point carefully in calculations.

340-N child is in a swing that is attached to a pair of ropes 1.90 m long. Find the gravitational potential energy of the child–Earth system relative to the child's lowest position at the following times.

(a) when the ropes are horizontal
 J

(b) when the ropes make a 34.0° angle with the vertical
 J

(c) when the child is at the bottom of the circular arc
 J

Find the gravitational potential energy of a child in a swing with ropes 1.90 m long. Explore scenarios where ropes are horizontal, at a 34° angle, and when the child is at the lowest point. Step-by-step calculations using U = mgh and trigonometry for height determination, ideal for Grades 10-12 physics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation