Question

You are designing an amusement park ride. A cart with two riders moves horizontally with speed \(v = 6.00\) m/s. You assume that the total mass of cart plus riders is 300 kg. The cart hits a light spring that is attached to a wall, momentarily comes to rest as the spring is compressed, and then regains speed as it moves back in the opposite direction. For the ride to be thrilling but safe, the maximum acceleration of the cart during this motion should be 3.00\(g\). Ignore friction. What is (a) the required force constant of the spring, (b) the maximum distance the spring will be compressed?

Short answer

(a) Spring constant \(k \approx 4905\, \text{N/m}\), (b) Maximum compression \(x \approx 1.53\, \text{m}\).

Step-by-step solution

Understand the Problem

First, identify what is being asked: we need to find the force constant of the spring (a) and the maximum compression of the spring (b). We have the following known values: velocity \(v = 6.00\, \text{m/s}\), mass \(m = 300\, \text{kg}\), and maximum acceleration \(a = 3.00\, g = 3.00 \times 9.81\, \text{m/s}^2\).

Use Energy Conservation to Relate Velocity and Spring Compression

Since the cart comes to rest, the initial kinetic energy is converted entirely to elastic potential energy of the spring. The initial kinetic energy \(KE = \frac{1}{2}mv^2\) and the potential energy stored in a compressed spring \(PE_s = \frac{1}{2}kx^2\), where \(k\) is the spring constant and \(x\) is the compression.Set these equal: \(\frac{1}{2}mv^2 = \frac{1}{2}kx^2\).

Derive Expression for Maximum Compression x

From \(\frac{1}{2}mv^2 = \frac{1}{2}kx^2\), simplify to find \(x\):\[ x = \sqrt{\frac{mv^2}{k}}\]

Determine Maximum Force Using Maximum Acceleration

Use Newton's second law where maximum force \(F_{\text{max}} = ma\). Therefore, \(F_{\text{max}} = 300 \times 3 \times 9.81\, \text{N}\).

Relate Force and Spring Compression to Find k

Hooke's law states that \(F_{\text{max}} = kx\). Substitute the expression for \(x\) from Step 3 into this equation to solve for \(k\):Substitute from Newton's law:\[ kx = 300 \times 3 \times 9.81 \]Solve for \(k\):\[ k = \frac{F_{\text{max}}}{x} = \frac{300 \times 3 \times 9.81}{\sqrt{\frac{300 \times 6^2}{k}}} \]

Calculate the Spring Constant k

To find \(k\), equate expressions for \(F_{\text{max}}\) using both \(kx\) and the derived formula for acceleration, where \(a = 3g\).Numerically calculate and simplify to find \(k\):\[ k = \frac{ma}{x} = \frac{300 \times 3 \times 9.81}{\frac{3600}{k}} = 300 \times 3 \times 9.81\, \text{N/m}\]

Calculate Maximum Compression x

Now use the expression derived in Step 3 to calculate \(x\) using the found value of \(k\). Substitute the numerical values to find \(x\):\[ x = \sqrt{\frac{300 \times 6^2}{k}} = \sqrt{\frac{10800}{k}} = 1.53 \text{m} \]

Confirm the Calculations

Re-evaluate your calculations for \(k\) and \(x\) for consistency with physical constraints: maximum force should not exceed the safety limit of 3g.

Key concepts

Spring Constant

In amusement park physics, the spring constant is a crucial parameter when designing rides that involve springs for safety and thrills. The spring constant, often symbolized as \( k \), tells us how stiff or flexible a spring is. It’s essentially a measure of how much force is needed to compress or extend the spring by a unit length.

To calculate the spring constant for the amusement ride described, we use the formula from Hooke's law, where the maximum force exerted by the spring when fully compressed is \( F = kx \). Here, \( x \) is the compression distance of the spring. In this scenario, the spring constant is derived by ensuring the ride remains thrilling yet safe, without exerting forces over the maximum allowable acceleration of 3g.

The spring constant helps determine the behavior of the cart during its motion. A high spring constant means the spring is quite stiff, requiring a significant force to compress, and resulting in less compression distance for a given force. Conversely, a low spring constant means the spring is more flexible.

Energy Conservation

In physics, energy conservation is a fundamental principle asserting that the total energy in a closed system remains constant over time. For the amusement park ride, this concept explains the motion as the kinetic energy of the moving cart converts into potential energy in the compressed spring, and then back into kinetic energy as the spring releases.

Initially, the cart possesses kinetic energy given by \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity. As the cart compresses the spring, the kinetic energy decreases while the spring potential energy, \( PE_s = \frac{1}{2}kx^2 \), increases until the cart momentarily stops. At that instant, all kinetic energy has changed into spring potential energy. This shift complies with energy conservation, highlighting that no energy is lost but rather transformed.

This principle ensures efficiency and safety in ride design, as engineers calculate and predict these energy transformations to maintain a thrilling yet controlled ride experience.

Newton's Second Law

Newton's Second Law of Motion is a core concept when analyzing movement in amusement park rides. It states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration, mathematically represented as \( F = ma \).

In the amusement park ride problem, Newton's second law aids in determining the forces involved as the cart interacts with the spring. Especially, it helps us know what maximum force corresponds to the set maximum acceleration. In this case, the maximum acceleration is specified to be 3 times the acceleration due to gravity, \( 3g \). Thus, the maximum force exerted during the ride is calculated by \( F_{\text{max}} = ma = 300 \times 3 \times 9.81 \).

Understanding this law ensures that the engineers can design components like the spring with precise specifications, ensuring maximum acceleration stays within safe limits, emphasizing both rider safety and ride excitement.

Hooke's Law

Hooke's Law is a staple in spring mechanics, indicating the relationship between the force applied to a spring and its displacement. It implies that the force needed to extend or compress a spring by some distance is proportional to that distance. Mathematically, it is expressed as \( F = kx \).

For the amusement park ride, Hooke's law allows us to relate the spring's compression to the force experienced by the riders. When the cart hits the spring and comes to rest, the maximum potential energy stored in the spring directly correlates to the kinetic energy the cart had initially. This conversion and the maximum force specification calculated through Newton's second law come together in Hooke's formulation.

By solving equations involving Hooke’s law, engineers determined both the spring constant and compression needed to achieve desired ride dynamics, ensuring compliance with safety standards while maximizing fun.