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Chapter 10: Problem 36

A cart with mass \(m\) is attached between two ideal springs, each with the same spring constant \(k\). Assume that the cart can oscillate without friction. (a) When the cart is displaced by a small distance \(x\) from its equilibrium position, what force magnitude acts on the cart? (b) What is the angular frequency, in terms of \(m, x,\) and \(k,\) for this cart? (W) tutorial: cart between springs) (IMAGE NOT COPY)

Short Answer

Expert verified
Answer: The force magnitude acting on the cart when it's displaced by a distance x from its equilibrium position is 0. The angular frequency for the cart in terms of m, x, and k is ω = √(2k/m).

Step by step solution

01

Understand Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position. Mathematically, the force by each spring is given by \(F = -kx\), where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position. Since the cart is attached between two ideal springs, the total force acting on the cart will be the sum of the forces from both springs.
02

Calculate the total force magnitude acting on the cart

The total force acting on the cart will be the sum of the forces from both springs. As the springs are connected in parallel, the force magnitudes are added together. The total force magnitude is given by the absolute value of the sum of the forces: \(|F_{total}| = |F_{1} + F_{2}| = |-kx + kx| = |0| = 0\). So, the force magnitude acting on the cart when it is displaced by a small distance x from its equilibrium position is 0.
03

Understanding the angular frequency formula

The angular frequency, denoted as \(\omega\), represents how fast the object oscillates in terms of radians per second. In the case of a mass-spring system, the angular frequency can be expressed as \(\omega = \sqrt{\frac{k_{eff}}{m}}\), where \(k_{eff}\) is the effective spring constant, and m is the mass of the oscillating object.
04

Determine the effective spring constant for the two-spring system

Since the springs are connected in parallel, their effective spring constant can be found by adding their individual spring constants. Since both springs have the same spring constant k, the effective spring constant is given by: \(k_{eff} = k + k = 2k\).
05

Calculate the angular frequency for the cart

Using the formula for angular frequency and the effective spring constant from Step 4, we can calculate the angular frequency for the cart in terms of m, x, and k: \(\omega = \sqrt{\frac{k_{eff}}{m}} = \sqrt{\frac{2k}{m}}\). So, the angular frequency for the cart is \(\omega = \sqrt{\frac{2k}{m}}\).

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