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 of the centripetal accelerations for the tips of the second hand and the minute hand, we need to understand the relationship between the periods and centripetal accelerations.
The centripetal acceleration (ac) of an object moving in a circle can be calculated using 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 can assume that both the second hand and the minute hand move in circles of different radii. Let's denote the radius of the second hand as r_second and the radius of the minute hand as r_minute.
Now, the velocity of an object moving in a circle can be calculated using the formula:
v = (2πr) / T
Where r is the radius of the circle and T is the period of the motion.
For the second hand, the radius is r_second, and the period is T_second. So, the velocity of the second hand can be written as:
v_second = (2πr_second) / T_second
Similarly, for the minute hand, the radius is r_minute, and the period is T_minute. So, the velocity of the minute hand can be written as:
v_minute = (2πr_minute) / T_minute
Now, let's substitute these expressions for velocity into the formula for centripetal acceleration:
ac_second = (v_second^2) / r_second
ac_minute = (v_minute^2) / r_minute
Substituting the expressions for velocity and rearranging, we get:
ac_second = [(2πr_second / T_second)^2] / r_second = (4π^2r_second) / T_second^2
ac_minute = [(2πr_minute / T_minute)^2] / r_minute = (4π^2r_minute) / T_minute^2
Finally, we can find the ratio of the centripetal accelerations:
ac_second / ac_minute = [(4π^2r_second) / T_second^2] / [(4π^2r_minute) / T_minute^2]
Simplifying this expression, we get:
ac_second / ac_minute = (r_second / T_second^2) / (r_minute / T_minute^2)
ac_second / ac_minute = (r_second * T_minute^2) / (r_minute * T_second^2)
Therefore, the algebraic expression for the ratio ac_second / ac_minute is (r_second * T_minute^2) / (r_minute * T_second^2).