Chapter 6, Problem 12Q

Analyze the motion of a simple swinging pendulum in terms of energy, (a) ignoring friction, and (b) taking friction into account. Explain why a grandfather clock has to be wound up.

To determine

The motion of a simple swinging pendulum in terms of energy ignoring friction.

Answer to Problem 12Q

Solution:The pendulum reaches at point "c" the same height $\left(h\right)$ as it was at point "a" from where it was released.

Explanation of Solution

Given:Motion in terms of energy for a simple swinging pendulumignoring friction.

If there is no friction mechanical energy can be written as:

  $E=K+U=0$  (1)

Or

  $K_f+U_f=K_i+U_i$  (2)

Equation (2) can be written as:

  $\frac{1}{2}m{v}_f^2+mg{h}_f=\frac{1}{2}m{v}_i^2+mg{h}_i$  (3)

Calculation:

A diagram of situation can be shown as follows:

  

Suppose that the pendulum is released from point a, its energy is entirely gravitational potential $\left(U_g\right)$ , when pendulum descends to point b, the gravitational potential energy $\left(U_g\right)$ is transformed into kinetic energy $\left(K\right)$ , therefore:

  $mgh=\frac{1}{2}m{v}_B^2$  (4)

Now, when pendulum ascends to point c the kinetic energy $\left(K\right)$ is transformed into gravitational potential energy $\left(U_g\right)$ reaching the same height $\left(h\right)$ as it was when it was released at point " a ", therefore:

  $\frac{1}{2}m{v}_B^2=mgh$  (5)

Conclusion: Comparing equations (4) and (5) the pendulum reaches at point " c " the same height $\left(h\right)$ as it was at point " a " from where it was released.

To determine

To DetermineThe motion of a simple swinging pendulum in terms of energy taking friction into account. Why a grandfather clock has to be wound up.

Answer to Problem 12Q

Solution:The total mechanical energy is not constant in the system, for this reason, the grandfather clock is adjusted with the help of springs to compensate for the loss of energy by friction.

Explanation of Solution

Given:Motion in terms of energy for a simple swinging pendulum taking friction into account.

Formula used: If there is friction then, equation (3) must be written as:

  $\frac{1}{2}m{v}_f^2+mg{h}_f=\frac{1}{2}m{v}_i^2+mg{h}_i+F_f.d$  (6)

In this case, part of the pendulum's kinetic and potential energy is transformed into thermal energy due to friction. As the oscillations advance, the kinetic and potential energy of the system is reduced until the pendulum stops completely. In the case of the grandfather clock, the same thing happens, for this reason, the watch is adjusted with the help of springs to compensate the energy lost by friction.

Conclusion:

The total mechanical energy is not constant in the system, for this reason, the grandfather clock is adjusted with the help of springs to compensate for the loss of energy by friction.