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 =
To find 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, we can use the formula for centripetal acceleration:
ac = 4π²r/T²,
where ac is the centripetal acceleration, r is the radius, and T is the period.
Let's assume that the radius of the second hand is rs, and the radius of the minute hand is rm.
For the second hand, the centripetal acceleration ac,second can be expressed as:
ac,second = 4π²rs/Ts².
For the minute hand, the centripetal acceleration ac,minute can be expressed as:
ac,minute = 4π²rm/Tm².
To find the ratio ac,second/ac,minute, we can divide ac,second by ac,minute:
ac,second/ac,minute = (4π²rs/Ts²)/(4π²rm/Tm²).
Simplifying the expression, we can cancel out the common factors:
ac,second/ac,minute = (rs/Ts²)/(rm/Tm²).
Expressing this ratio in terms of the periods Tsecond and Tminute, we have:
ac,second/ac,minute = (rsTm²)/(rmTs²).