- WELCOME TO A SHORT LESSON ON HOOKE'S LAW.
HOOKE'S LAW IS A PRINCIPLE OF PHYSICS THAT STATES THE FORCE F
NEEDED TO STRETCH OR COMPRESS A SPRING BY SOME DISTANCE, S,
IS PROPORTIONAL TO THAT DISTANCE.
AND SINCE F IS PROPORTIONAL TO S,
WE CAN USE THE DIRECT VARIATION EQUATION F = K x S
TO REPRESENT THIS RELATIONSHIP,
WHERE F IS THE FORCE,
K IS THE SPRING CONSTANT
AND S IS THE DISTANCE THE SPRING IS STRETCHED.
IF WE LOOK AT THE DIAGRAM BELOW,
IF WE START ON THE LEFT HERE,
THIS IS A SPRING THAT'S NOT STRETCHED
AND THEN WE ATTACH A WEIGHT TO IT OR A FORCE
AND WE CAN SEE THE SPRING IS STRETCHED S UNITS.
AND THE RELATIONSHIP BETWEEN THE FORCE F AND THE DISTANCE S
IS MODELED BY THIS EQUATION HERE, F = K x S,
WITH THE CONSTANT K WILL ALWAYS BE THE SAME FOR A GIVEN SPRING.
LET'S TAKE A LOOK AT AN EXAMPLE.
IF 10 POUNDS STRETCHES A SPRING HALF A FOOT,
WHAT IS THE SPRING CONSTANT
AND THEN HOW MANY INCHES WILL 6 POUNDS STRETCH THE SPRING?
SO, WE FIRST NEED TO RECOGNIZE THAT OUR FORCE F
IS GOING TO BE 10 POUNDS
AND SINCE THE SPRING IS STRETCHED HALF A FOOT,
S IS 1/2 OR 1/2 OF A FOOT.
BUT BECAUSE THIS SECOND QUESTION ASKS US TO FIND
HOW MANY INCHES 6 POUNDS WILL STRETCH THE SPRING,
LET'S GO AHEAD AND WRITE 1/2 A FOOT AS 6 INCHES.
AND NOW, USING THE FORMULA,
F = K x S, WE'LL PERFORM SUBSTITUTION FOR F AND S
AND SOLVE FOR K, WHICH WILL BE OUR SPRING CONSTANT.
SO, WE'D HAVE 10 = K TIMES 6
TO SOLVE FOR K, WE WOULD DIVIDE BOTH SIDES BY 6.
SO, WE CAN SEE K = 10/6.
LET'S GO AHEAD AND SIMPLIFY THIS.
THERE'S A COMMON FACTOR OF 2 BETWEEN 10 AND 6.
SO, OUR SPRING CONSTANT IS 5/3.
SO, THE ANSWER TO THE SECOND QUESTION,
HOW MANY INCHES WILL 6 POUNDS STRETCH THE SPRING,
WE CAN NOW USE THE DIRECT VARIATION EQUATION,
F = 5/3 x S,
TO ANSWER THIS QUESTION, WE WANT TO FIND S WHEN F = 6 POUNDS.
SO, OUR EQUATION WOULD BE 6 = 5/3 x S.
TO SOLVE FOR S, WE'LL MULTIPLY BY THE RECIPROCAL OF 5/3
ON BOTH SIDES OF THE EQUATION, WHICH WOULD BE 3/5.
SO, WE'LL MULTIPLY THE LEFT SIDE BY 3/5
AND THE RIGHT SIDE BY 3/5.
LET'S GO AHEAD AND PUT 6/1.
NOTICE ON THE RIGHT SIDE, 3/3 SIMPLIFIES TO 1 AND SO DOES 5/5.
SO, WE HAVE S EQUALS.
NOTHING SIMPLIFIES HERE.
WE HAVE 18/5.
REMEMBER THIS IS THE DISTANCE THAT THE SPRING WILL STRETCH
IN INCHES.
SO, S = 18/5 INCHES.
LET'S CONVERT THIS TO A MIXED NUMBER, 18 DIVIDED BY 5.
THERE ARE THREE 5s IN 18, 3 x 5 IS 15, SUBTRACT.
WE HAVE A REMAINDER OF 3,
WHICH MEANS WE CAN ALSO EXPRESS THIS AS 3 AND 3/5 INCHES.
AND SINCE 3/5 = 6/10, WE COULD ALSO EXPRESS S AS A DECIMAL.
WE COULD ALSO SAY S = 3 AND 6/10 OR 3.6 INCHES.
ALL OF THESE LENGTHS ARE EQUIVALENT.
LET'S DETERMINE IF OUR ANSWER MAKES SENSE.
NOTICE WE WERE TOLD THAT A 10-POUND WEIGHT
STRETCHES A SPRING 6 INCHES.
SO, IF WE ATTACH A 6-POUND WEIGHT, WHICH IS LESS,
NOTICE HOW IT DOES STRETCH THE SPRING LESS.
IT ONLY STRETCHES 3 AND 3/5 INCHES,
WHICH DOES MAKE SENSE.
I HOPE YOU FOUND THIS HELPFUL.