What is the algebraic expression for the ratio ac,second/ac,minute of the centripetal accelerations for the tips of the second hand and the minute hand? Express your answer in terms of the periods Tsecond and Tminute. (Answer using T_m to be the period of the second hand and T_s to be period of the minute hand.)

ac,second/ac,minute =

(d) What is the ratio ac,second/ac,minute of the centripetal accelerations for the tips of the second hand and the minute hand?
ac,second/ac,minute =

Let L be the length of a clock's hand and T be its period of rotation. The acceleration of the tip is

a = V^2/L = (2 pi L/T)^2/T = 4 pi^2 L^2/T

Note the inverse dependence upon T. If both hands had the same length L, the a of the second hand would be 60 times the a of the minute hand.

ac,second/ac,minute = T_minute/T_second = 60

To find the algebraic expression for the ratio ac,second/ac,minute, we need to start by understanding the concept of centripetal acceleration and how it relates to the periods Tsecond and Tminute.

Centripetal acceleration (ac) is the acceleration experienced by an object moving in a circular path. It is given by the formula ac = (v^2) / r, where v is the velocity of the object and r is the radius of the circular path.

In this case, we are interested in the centripetal accelerations of the tips of the second hand and the minute hand of a clock. The periods Tsecond and Tminute represent the time it takes for the second hand and the minute hand to complete one full revolution, respectively.

The formula for the period of an object moving in a circle is T = 2πr / v, where T is the period, r is the radius, and v is the velocity.

Let's assume that the radius of the circular path for both the second hand and the minute hand is the same. This is a reasonable assumption since the hands of a clock typically rotate around a fixed center.

Now, to find the ratio ac,second / ac,minute, we can express the velocities and radii in terms of the periods Tsecond and Tminute.

The velocity can be calculated using v = 2πr / T. Since the radius is the same for both cases, we can write vsecond = 2πr / Tsecond and vminute = 2πr / Tminute.

Substituting these velocities into the formula for centripetal acceleration, we get ac,second = (vsecond^2) / r = (4π^2r^2) / (Tsecond^2) and ac,minute = (vminute^2) / r = (4π^2r^2) / (Tminute^2).

Now, to find the ratio ac,second / ac,minute, we can divide ac,second by ac,minute:

ac,second / ac,minute = [(4π^2r^2) / (Tsecond^2)] / [(4π^2r^2) / (Tminute^2)]

Simplifying this expression, we can cancel out the common terms:

ac,second / ac,minute = (Tminute^2) / (Tsecond^2)

So, the algebraic expression for the ratio ac,second / ac,minute is (Tminute^2) / (Tsecond^2).

Therefore, the answer to the question is ac,second/ac,minute = (Tminute^2) / (Tsecond^2).