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Chapter 5: Problem 24

A \(3.50 \times 10^{2}-\mathrm{N}\) child is in a swing that is attached to ropes \(1.75 \mathrm{~m}\) long. Find the gravitational potential energy associated with the child relative to her lowest position when (a) the ropes are horizontal, (b) the ropes make a \(30.0^{\circ}\) angle with the vertical, and (c) the child is at the bouom of the circular arc.

Short Answer

Expert verified
The gravitational potential energies for the respective scenarios are approximately \(615.615 J\), depending on the calculated height for the case when ropes make a \(30.0^{\circ}\) angle with the vertical and \(0 J\) when the child is at the bottom of the circular arc.

Step by step solution

01

Calculate Child's Mass

First, calculate the mass of the child using the provided weight and the relation between weight (W), mass (m), and gravity (g): \(W = mg\). The weight of the child is given as \(350 N\). From this, the mass of the child can be calculated as \(m = W/g = 350 N / 9.8 m/s^{2} \approx 35.7 kg\).
02

Calculate Gravitational Potential Energy When Ropes are Horizontal

In this situation, the length of the ropes corresponds to the height (h). Using the Gravitational Potential Energy equation mentioned earlier, \(PE_{g} = mgh = 35.7 kg \times 9.8 m/s^{2} \times 1.75 m \approx 615.615 J\).
03

Calculate Gravitational Potential Energy When Ropes Make a \(30.0^{\circ}\) Angle with Vertical

When the ropes make a 30-degree angle with the vertical, the height h is calculated as: \(h = 1.75 m * (1 - cos30^{\circ})\). The gravitational potential energy is then: \(PE_{g} = mgh = 35.7 kg \times 9.8 m/s^{2} \times h (calculated before)\).
04

Calculate Gravitational Potential Energy When Child is at Bottom of Circular Arc

When the child is at the bottom of the circular arc, the height is 0 since the reference point is her lowest position. Therefore, the gravitational potential energy is also zero: \(PE_{g} = mgh = 35.7 kg * 9.8 m/s^{2} * 0 = 0 J\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Weight and Mass Conversion
Before diving into calculating gravitational potential energy, it's important to understand the relationship between weight and mass. In physics, weight is the force exerted on an object due to gravity. It's calculated as the product of mass and the acceleration due to gravity. This relationship is described by the formula:
  • Weight ( \(W\) ) = Mass ( \(m\) ) x Gravity ( \(g\) )
For our exercise, the weight of the child is given as \(350 N\). The acceleration due to gravity on Earth, \(g\), is approximately \(9.8 m/s^{2}\).
To find the mass, rearrange the formula to give mass (\(m = W/g\)). Thus, the child's mass can be calculated as approximately \(35.7 kg\).
Understanding this conversion is crucial because knowing the mass allows us to compute other forces and energies, such as gravitational potential energy.
Rope Length and Angle
In scenarios involving swings, pendulums, or ropes, knowing the length of the rope and the angles it forms is essential. This knowledge helps in calculating the height or displacement of the object from its lowest point. The height needs to be determined to find the gravitational potential energy, \(PE_{g}\).
When the swing makes a \(30^{\circ}\) angle with the vertical, it's vital to understand how the cosine function helps in finding the height, \(h\). The formula to find the height when the ropes are at any angle \(\theta\) with the vertical is:
  • \(h = L\ (1 - \cos\theta)\)
where \(L\) is the length of the rope.
For the exercise, when the angle is \(30^{\circ}\) and \(L = 1.75 m\), substitute to find \(h\).
This approach aids in calculating gravitational potential energy accurately in various positions of the swing.
Circular Motion
Objects in a circular or arc motion encounter unique mechanical situations due to their paths. For a simple swing system, as the child moves in a circular arc, different points along the swing's path represent different potential energies relative to her lowest swing position.
At the highest point on either side, the swing gains maximum gravitational potential energy due to its height from the lowest point. In our specific exercise, it is calculated when the ropes are horizontal, making the height equal to the rope length:
  • \(PE_{g} = 35.7 kg \times 9.8 m/s^{2} \times 1.75 m\)
Conversely, when the child is at the bottom of the arc, her height is zero, and thus, the gravitational potential energy is zero. This dynamic change in energy illustrates how gravitational potential energy varies with motion in a gravitational field.
Understanding these changes is integral in analyzing systems involving swings or motion paths that work against gravity.

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Most popular questions from this chapter

Tarzan swings on a \(30.0\) -m-long vine initially inclined at an angle of \(37.0^{\circ}\) with the vertical. What is his speed at the bottom of the swing (a) if he starts from rest? (b) If he pushes off with a speed of \(4.00 \mathrm{~m} / \mathrm{s}\) ?

A light spring with spring constant \(1.20 \times 10^{3} \mathrm{~N} / \mathrm{m}\) hangs from an elevated support. From its lower end hangs a second light spring, which has spring constant \(1.80 \times 10^{3} \mathrm{~N} / \mathrm{m}\). A \(1.50-\mathrm{kg}\) object hangs at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as being in series. Hint: Consider the forces on each spring separately.

An archer pulls her bowstring back \(0.400 \mathrm{~m}\) by exerting a force that increases uniformly from zero to \(230 \mathrm{~N}\). (a) What is the equivalent spring constant of the bow? (b) How much work does the archer do in pulling the bow?

Be The chin-up is one exercise that can be used to strengthen the biceps muscle. This muscle can exert a force of approximately \(800 \mathrm{~N}\) as it contracts a distance of \(7.5 \mathrm{~cm}\) in a \(75-\mathrm{kg}\) male. \({ }^{3}\) How much work ean the biceps muscles (one in each arm) perform in a single contraction? Compare this amount of work with the energy required to lift a \(75-\mathrm{kg}\) person \(40 \mathrm{~cm}\) in performing a chin-up. Do you think the biceps muscle is the only muscle involved in performing a chin- up?

The electric motor of a model train accelerates the train from rest to \(0.620 \mathrm{~m} / \mathrm{s}\) in \(21.0 \mathrm{~ms}\), The total mass of the train is \(875 \mathrm{~g} .\) Find the average power delivered to the train during its acceleration.
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