Final answer:
The second hand of the wall clock is 22.0 cm long. For the practice problems, the angular velocity vector points towards the observer, arc length can be found using the angle between clock hands and radius, and the distance from the axis of rotation for a specific radial acceleration can be determined through the formula involving angular velocity and radius.
Step-by-step explanation:
The length of the second hand on the wall clock is explicitly stated as 22.0 cm. Since this is a straightforward fact provided in the question, no further calculation is needed.
Regarding the practice problems provided, here's a brief explanation for each:
- For the clock mounted on the wall, the direction of the angular velocity vector of the second hand is perpendicular to the plane of the clock face, following the right-hand rule, it points towards the observer if the clock is moving in the customary clockwise direction.
- The arc length between the hour hand and the minute hand at 10:00 a.m. with a radius of 0.2 m can be found by calculating the angle between the hands and then using the formula for arc length: θr, where θ is the angle in radians and r is the radius.
- A point on the second hand with a radial acceleration of 0.1cm/s² is at a distance from the axis of rotation that can be found using the formula a = ω²r, where a is the radial acceleration, ω is the angular velocity, and r is the radius.
- Changing the length of a clock's pendulum by 1.000% affects the period of the pendulum, hence the timing of the clock. Using the precise percentage change and the relation between the pendulum length and the period, one can calculate the time the clock will read 24 hours later. Since the period of the pendulum is proportional to the square root of its length, a longer pendulum will result in a clock that runs slower, and a shorter one will cause it to run faster.