MATHEMATICS: Ratio, Proportion, Variation - FREEFALL, DISPLACEMENT, RESISTOR POWER and CURRENT, PENDULUM OSCILLATIONS and LENGTH

1. DISTANCE TRAVELLED is directly proportional to the SQUARE OF THE TIME OF TRAVEL (freefall, freely falling bodies)


example:

If a body falls 200 meters in 8 seconds, How far will it fall in 10 seconds?


given:

S1 = 200 m

t1 = 8 sec

t2 = 10 sec


solution:


S = kt^2

k = S/t^2

S1/(t1)^2 = S2/(t2)^2

200/(8)^2 = S2/(10)^2

200/64 = S2/100

S2 = 200 * 100/64

S2 = 312.5 m




2. POWER IN A RESISTOR is directly proportional to the SQUARE OF THE CURRENT


example:

How many times the current in an electric heating coil be increased such that the power consumed will be 4 times.


given:

P1 = x

P2 = 4x


solution:


P = kI^2

k = P/I^2

P1/(I1)^2 = P2/(I2)^2

x/(I1)^2 = 4x/(I2)^2

(I2)^2 = (I1)^2 * 4x/x

(I2)^2/(I1)^2 = 4


if I1 = 7 amp


(I2)^2/(7)^2 = 4

(I2)^2/49 = 4

(I2)^2 = 4 * 49

(I2)^2 = 196

(I2) = sqrt (196)

I2 = 14 amp ---> twice of I1


therefore, if the POWER consumed is 4 times, the CURRENT will be doubled




3. PENDULUM OSCILLATIONS inversely proportional to SQUARE ROOT OF LENGTH OF PENDULUM


example:

A pendulum of 12 in length oscillates at 2 oscillations per second. If the length is increased twice, that is, 24 in., how many oscillations does the pendulum produces?


given:

n1 = 2 oscillations/sec

L1 = 12 in

L2 = 24 in


find:

n2 = oscillations if the length is doubled



solution:


n = k/sqrt(L)

k = n * sqrt(L)

n1 * sqrt(L1) = n2 * sqrt(L2) 

2 * sqrt(12) = n2 * sqrt(24)

n2 = 2 * sqrt(12)/sqrt(24)

n2 = 2 * 3.464/4.899

n2 = 2 * 0.707

n2 = 1.4 oscillations/second