When a force is applied to an object, it responds in a manner that is dependent upon its molecular composition. Some objects are easily ( and permanently ) deformed by forces applied to them. In other cases, an object will recoil and return to its original form after being temporarily deformed by a force. Objects such as these are elastic. The topic of elasticity in physics can be addressed in part by Hooke’s Law, which states that an elastic object will be stretched or compressed by some distance ( s ) in proportion to an applied force ( F ):
There are limits, however, to how much force can be applied to a springy system before it too becomes permanently deformed. The proportional relationship between applied force and displacement gives rise to the need for a constant of proportionality ( k ) that can be applied to each springlike system as is appropriate:
F = ks
( F / s ) = k
The spring constant, a.k.a. the elastic force constant, will give rise to a relatively steep force-displacement slope when a springy system is stiff, whereas less-stiff systems give rise to graphs with more modest slopes as diagrammed above. The spring constant is expressed in terms of Newton’s per meter ( N / m ).
Among other things, measurements of the extent to which a spring is linearly displaced by a force enables spring scales to be designed with convenience. Weight is is force ( Fw ), and near the surface of the earth, it acts on masses in proportion to the gravitational constant of acceleration ( g ):
Fw = mg
When a spring scale is being calibrated, a quantity of mass needed to fully extend its inner spring is attached to it. Next, the extent to which the spring is displaced is recorded. This enables a spring constant for the system to be determined:
Once the spring constant has been recorded, the mass of a suspended object can easily be determined by multiplying ( k ) by the extent to which the object is displaced. This yields a value of force that can be divided by ( g ) to determine that value of the suspended mass:
Q: During calibration, a spring scale is maximally displaced 500.0 cm by a 500.0 g suspended mass. The spring constant is determined as follows:
Fw = mg
Fw = ( 0.500 kg )( 9.86 m/s2 )
Fw = 4.93 N
( Fw / s ) = k
k = ( 4.93 N / 0.500 m ) = 9.86 N / m
What is the mass of an object that displaces the scale by 3.00 cm?
A: Fs = ks
Fs = ( 9.86 N / m )( 0.300 m )
Fs = 2.96 N
( Fs / g ) = m
( 2.96 N / 9.86 m / s2 ) = 0.3 kg
RENAISSANCE PHYSICS 