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Chapter 13: Problem 49

Derive the period of a simple pendulum by considering the horizontal displacement \(x\) and the force acting on the bob, rather than the angular displacement and torque.

Short Answer

Expert verified
The period of a simple pendulum is given by \(T = 2\pi \sqrt{\frac{L}{g}}\).

Step by step solution

01

Identify the parameters involved

To start, we need to identify the parameters involved. A simple pendulum consists of a mass \(m\) (the pendulum's bob) attached to a lightweight cord of length \(L\), which swings back and forth under the influence of gravity. The displacement \(x\) at any time \(t\) can be expressed as \(x = L sin(\theta)\), where \(\theta\) is the angular displacement.
02

Write down the force acting on the bob

The force acting on the bob is the component of the gravitational force in the direction of motion. Therefore, it is given by \(F = -mg sin(\theta)\). Using the small-angle approximation \(sin(\theta) \approx \theta\), when the pendulum is at small displacements, this becomes \(F = -mg \frac{x}{L}\).
03

Apply Newton's Second Law of motion

Newton鈥檚 Second Law states that the force applied to an object equals its mass times acceleration \(F = ma\). In our case, the acceleration is the second derivative of displacement \(x\) with respect to time \(t\). Hence, we have an equation \(-mg \frac{x}{L} = m \frac{d^2x}{dt^2}\). The mass \(m\) cancels out.
04

Solve the differential equation

Now we have to solve the ordinary differential equation. Rearranging gives \(\frac{d^2x}{dt^2} + g\frac{x}{L} = 0\). This is a simple harmonic motion equation with a general solution of \(x(t) = A cos(\omega t + \phi)\), where \(A\) and \(\phi\) are determined by initial conditions, and \(\omega = \sqrt{g/L}\) is the angular frequency.
05

Compute the period of the pendulum

The period \(T\) of the pendulum is the time it takes for one complete cycle, which is \(\frac{2\pi}{\omega}\). Thus, \(T = 2\pi \sqrt{\frac{L}{g}}\). This is the same result obtained when using the angular displacement and torque. It confirms the physical law: the period of a simple pendulum depends only on its length and the acceleration due to gravity.

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Most popular questions from this chapter

A simple model for a variable star considers that the outer layer of the star is subject to two forces: the inward force of gravity and the outward force due to gas pressure. As a result, Newton's law for the star's outer layer reads \(m d^{2} r / d t^{2}=4 \pi r^{2} p-G M m / r^{2} .\) Here \(m\) is the mass of the outer layer, \(M\) is the total mass of the star, \(r\) is the star's radius, and \(p\) is the pressure. (a) Use this equation to show that the star's equilibrium pressure and radius are related by \(p_{0}=G M m / 4 \pi r_{0}^{4},\) where the subscript 0 represents equilibrium values. (b) As you'll learn in Chapter 18 , gas pressure and volume \(V\left(=\frac{4}{3} \pi r^{3}\right)\) are related by \(p V^{3 / 3}=p_{0} V_{0}^{5 / 3}\) (this is for an adiabatic process, a good approximation here, and the exponent \(5 / 3\) reflects the ionized gas that makes up the star). Let \(x=r-r_{0}\) be the displacement of the star's surface from equilibrium. Use the binomial approximation (Appendix A) to show that, when \(x\) is small compared with \(r,\) the righthand side of the above equation can be written \(-\left(G M m / r_{0}^{3}\right) x\) (c) since \(r\) and \(x\) differ only by a constant, the term \(d^{2}\) r/dt \(^{2}\) in the equation above can also be written \(d^{2} x / d t^{2} .\) Make this substitution, along with substituting the result of part (b) for the right- hand side, and compare your result with Equations 13.2 and 13.7 to find an expression for the oscillation period of the star. (d) What does your simple model predict for the period of the variable star Delta Cephei, with radius 44.5 times that of the Sun and mass of 4.5 Sun masses? (Your answer overestimates the actual period by a factor of about \(3,\) both because of oversimplified physics and because changes in the star's radius are too large for the assumption of a linear restoring force.)

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The protein dyeing powers the flagella that propel some unicellular organisms. Biophysicists have found that dyeing is intrinsically oscillatory, and that it exerts peak forces of about \(1.0 \mathrm{pN}\) when it attaches to structures called micro-tubules. The resulting oscillations have amplitude \(15 \mathrm{nm}\). (a) If this system is modeled as a mass-spring system, what's the associated spring constant? (b) If the oscillation frequency is \(70 \mathrm{Hz}\), what's the effective mass?

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