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Chapter 6: Problem 100

Two springs with spring constants \(k_{1}\) and \(k_{2}\) are connected in series. (a) What is the effective spring constant of the combination? (b) If a hanging object attached to the combination is displaced by \(4.0 \mathrm{cm}\) from the relaxed position, what is the potential energy stored in the spring for \(k_{1}=5.0 \mathrm{N} / \mathrm{cm}\) and $k_{2}=3.0 \mathrm{N} / \mathrm{cm} ?$ [See Problem \(83(\mathrm{a}) .]\)

Short Answer

Expert verified
Answer: The potential energy stored in the springs is \(15 \, J\).

Step by step solution

01

Determine the effective spring constant of springs in series

Consider two springs with spring constants \(k_1\) and \(k_2\) connected in series. The effective spring constant \(K\) can be calculated using the formula: \[ \frac{1}{K} = \frac{1}{k_1} + \frac{1}{k_2} \] Rearrange the equation to solve for \(K\): \[ K = \frac{k_1k_2}{k_1 + k_2} \] We will use this equation to find the effective spring constant of the combination.
02

Calculate the effective spring constant for the given values

Now, we will substitute the given values \(k_1 = 5.0 \, N/cm\) and \(k_2 = 3.0 \, N/cm\) into the equation: \[ K = \frac{(5.0)(3.0)}{5.0 + 3.0} = \frac{15}{8} \, N/cm \] The effective spring constant of the springs in series is \(\frac{15}{8} \, N/cm\).
03

Calculate the potential energy stored in the spring for the given displacement

The potential energy \(U\) stored in a spring can be calculated using the formula: \[ U = \frac{1}{2} K x^{2} \] where \(K\) is the effective spring constant and \(x\) is the displacement from the relaxed position. We will use the effective spring constant found in step 2 and the given displacement of \(4.0 \, cm\) to calculate the potential energy stored in the spring.
04

Calculate the potential energy stored in the spring for the given displacement and effective spring constant

Now, substitute the effective spring constant \(K = \frac{15}{8} \, N/cm\) and the displacement \(x = 4.0 \, cm\) into the formula for the potential energy: \[ U = \frac{1}{2} \times \frac{15}{8} \times (4.0)^{2} = \frac{15}{8} \times 8 = 15 \, J \] The potential energy stored in the spring is \(15 \, J\).

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Most popular questions from this chapter

Yosemite Falls in California is about 740 m high. (a) What average power would it take for a 70 -kg person to hike up to the top of Yosemite Falls in $1.5 \mathrm{h} ?\( (b) The human body is about \)25 \%$ efficient at converting chemical energy to mechanical energy. How much chemical energy is used in this hike? (c) One food Calorie is equal to \(4.186 \times 10^{3} \mathrm{J} .\) How many Calories of food energy would a person use in this hike?
Hilda holds a gardening book of weight \(10 \mathrm{N}\) at a height of $1.0 \mathrm{m}\( above her patio for \)50 \mathrm{s}$. How much work does she do on the book during that 50 s?
A block of wood is compressed 2.0 nm when inward forces of magnitude $120 \mathrm{N}$ are applied to it on two opposite sides. (a) Assuming Hooke's law holds, what is the effective spring constant of the block? (b) Assuming Hooke's law still holds, how much is the same block compressed by inward forces of magnitude \(480 \mathrm{N} ?\) (c) How much work is done by the applied forces during the compression of part (b)?
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