Question

A tower outside the Houses of Parliament in London has a famous clock commonly referred to as Big Ben, the name of its 13 -ton chiming bell. The hour hand of each clock face is \(2.7 \mathrm{m}\) long and has a mass of \(60.0 \mathrm{kg}\) Assume the hour hand to be a uniform rod attached at one end. (a) What is the torque on the clock mechanism due to the weight of one of the four hour hands when the clock strikes noon? The axis of rotation is perpendicular to a clock face and through the center of the clock. (b) What is the torque due to the weight of one hour hand about the same axis when the clock tolls 9: 00 A.M.?

Short answer

Answer: The torque on the clock mechanism due to the weight of one of the hour hands at noon is 794.61 Nm, and at 9:00 AM, it is 562.925 Nm.

Step-by-step solution

Finding the weight of the hour hand

First, we find the weight of an hour hand, which is given by the following formula: \(Weight (W) = Mass (m) × g\), where \(g\) is the acceleration due to gravity (\(9.81 m/s^2\)) Given mass \(m = 60.0 kg\) \(W = 60.0 kg * 9.81 m/s^2 = 588.6 N\)

Torque at noon

At noon, the force of gravity acts at the center of the hour hand which is halway along the length. Therefore, the perpendicular distance to the axis of rotation is half the length of the hand (\(2.7 m/2 = 1.35 m\)). Now, we can find the torque: \(Torque (\tau_{noon}) = Weight (W) × \text{perpendicular distance}\) \(\tau_{noon}= 588.6 N * 1.35 m = 794.61 Nm\)

Torque at 9:00 AM

At 9:00 AM, the configuration of the clock changes and the perpendicular distance from the center of mass of the hour hand can be found using geometry. The hour hand forms a 45-degree angle with the vertical and the distance will be \(\frac{2.7m}{2} \cos(45^\circ)\). Using trigonometry: Perpendicular distance \(= \frac{2.7m}{2} \cos(45^\circ) = \frac{2.7m}{2} \frac{1}{\sqrt{2}} = \frac{2.7m}{\sqrt{8}} = 0.9564m\) Now, we can find the torque: \(Torque (\tau_{9AM}) = Weight (W) × \text{perpendicular distance}\) \(\tau_{9AM} = 588.6 N * 0.9564 m = 562.925 Nm\)

Summary of the Results

a) The torque on the clock mechanism due to the weight of one of the hour hands when the clock strikes noon is \(794.61 Nm\). b) The torque due to the weight of one hour hand about the same axis when the clock tolls 9:00 AM is \(562.925 Nm\).