A block (mass 0.5 kg) is hanging from the end of a spring. The spring is observed to
stretch by 20 cm in order to come to equilibrium.
a) What is the spring constant?
F=-kx
F=mg=.5(9.8)=4.9 N and x=-.2meters
-4.9/-.2meters=24.5 =k
should the constant be negative or positive because I said that x is negative? Is that right?
b) From the equilibrium position the block is now stretched by 10 cm and released from rest.
What is the period of the oscillations?
so w=sqrt(k/m)=sqrt(24.5/.5)=7 rad/s
f=w/2pi=1.11 Hz
T=1/f=0.901 s
a) correct value. Spring constants are always positive.
b) also correct, but I get 0.896 for T, if I keep three significant figures. Since only one siginificant fgure is justified, 0.9 s is OK for the period.
a) The spring constant, represented by the symbol "k," is the constant that relates the force exerted by a spring to the displacement of the spring from its equilibrium position. In this case, you are given the mass of the block, which is 0.5 kg, and the displacement of the spring when it comes to equilibrium, which is 20 cm or 0.2 meters.
To find the spring constant, you can use Hooke's Law, which states that the force exerted by a spring is equal to the negative product of the spring constant and the displacement from the equilibrium position.
F = -kx
In this equation, F represents the force exerted by the spring, which is also equal to the weight of the block hanging from the spring. Since the block is hanging vertically, its weight is given by the equation:
F = mg
where m is the mass of the block and g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Substituting these values into Hooke's Law, you have:
mg = -kx
Solving for k:
k = -mg/x = -(0.5)(9.8)/(-0.2) = 24.5 N/m
The spring constant should be positive because you are dealing with magnitudes. The negative sign in the formula simply indicates direction; it tells you that the force exerted by the spring is in the opposite direction of the displacement.
b) To find the period of the oscillations, you can use the equation:
T = 2π/w
where T is the period and w is the angular frequency. The angular frequency is given by:
w = √(k/m)
From part (a), you already know the spring constant, k, and the mass, m. Substituting these values into the equation for angular frequency, you get:
w = √(24.5/0.5) = 7 rad/s
Finally, substituting the angular frequency into the equation for period, you find:
T = 2π/7 ≈ 0.901 s