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Chapter 7: Problem 46

The maximum speed of the pendulum bob in a grandfather clock is \(0.55 \mathrm{m} / \mathrm{s} .\) If the pendulum makes a maximum angle of \(8.0^{\circ}\) with the vertical, what's the pendulum's length?

Short Answer

Expert verified
The length of the pendulum is approximately 0.99 m.

Step by step solution

01

Identify the pendulum鈥檚 maximum kinetic energy

The pendulum鈥檚 maximum kinetic energy (K.E.) is when it's at its maximum speed, which is given as \(0.55 m/s\). The formula for kinetic energy is \(K.E. = \frac{1}{2} m v^{2}\), where \(m\) is the mass of the object and \(v\) is its velocity. However, in this case, we will consider only the velocity as the mass \(m\) will eventually cancel out.
02

Consider the pendulum鈥檚 potential energy

The pendulum鈥檚 maximum potential energy (P.E) is at the peak of its swing, which is when it has swung to its maximum angle of \(8.0掳\). The formula for gravitational potential energy is \(P.E = m g h\), where \(m\) is the mass, \(g\) is acceleration due to gravity, and \(h\) is the height. The height \(h\) can be expressed in terms of the pendulum's length \(L\) and the maximum angle \(\theta\) it makes with the vertical. Therefore, \(h = L - L cos(\theta)\). Since energy must be conserved, the maximum kinetic and potential energy are equal which gives us \( \frac{1}{2} m v^{2} = m g (L - L cos(\theta))\).
03

Solve for the Pendulum鈥檚 Length

Rearranging for L gives \(L = \frac{v^2}{2g(1-cos(\theta))}\). Substituting the values \(v = 0.55 m/s\), \(\theta = 8.0掳 = 8.0 \times \frac{蟺}{180} rad\), and \(g = 9.8 m/s^2\), and performing the calculations gives L. Note that the mass \(m\) cancels out, hence it's not needed

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