Clocks and Calendars

The time period of a pendulum is T = 2 * pi * sqrt(L/g). The change in time is given by delta T = (1/2) * alpha * delta theta * T. For one day (86400 seconds), the loss is (1/2) * (2 * 10^-5) * (35 - 20) * 86400 = 10^-5 * 15 * 86400 = 12.96 seconds.

Hands of a clock are in a straight line but not together when they are 180 degrees apart. This happens once every hour, except between 5 and 7, where it happens only once at 6 o'clock. Thus, it occurs 22 times in 24 hours.

Angle = |30H - 5.5M|. H=4, M=20. Angle = |30*4 - 5.5*20| = |120 - 110| = 10 degrees.

The hands of a clock overlap every 65 and 5/11 minutes. In a 12-hour period, they overlap 11 times. Since the interval is from noon to midnight (12 hours), the hands overlap 11 times.

The watch gains 10 minutes in 24 hours (1440 minutes). The gain per minute is 10/1440 = 1/144. From 9 pm to 2 am is 5 hours (300 minutes). The gain is 300 / 144 = 2.0833 minutes, which is 2 minutes and 5 seconds.

At 5 o'clock, the minute hand is at 12 and the hour hand is at 5 (25 minutes apart). For a 90-degree angle, the hands must be 15 minutes apart. The minute hand must gain 25 - 15 = 10 minutes. Time = 10 * (60/55) = 10 * 12/11 = 120/11 = 10 10/11 minutes past 5.

At 2:25, the minute hand is at 5 (25 minutes). The hour hand is at 2 + 25/60 = 2 + 5/12 = 29/12. The angle of the minute hand is 25 * 6 = 150 degrees. The angle of the hour hand is (29/12) * 30 = 72.5 degrees. The difference is 150 - 72.5 = 77.5 degrees.