A light spring with a spring constant of 15.2 N/m rests vertically on the bottom of a large beaker of water, as shown in (a) below. A 5.26 10-3 kg block of wood with a density of 636.0 kg/m3 is connected to the spring, and the mass-spring system is allowed to come to static equilibrium, as shown in (b) below. How much does the spring stretch?
To find out how much the spring stretches, we need to consider the equilibrium of forces acting on the system.
First, let's calculate the buoyant force acting on the block of wood submerged in water.
The density of water is constant at 1000 kg/m^3. The block of wood's density is given as 636.0 kg/m^3.
Buoyant force (F_b) = density of fluid × gravity × volume of the immersed object
The volume of the block (V) can be found using its mass and density: V = mass / density
Given that the mass of the block is 5.26 × 10^(-3) kg, and its density is 636.0 kg/m^3, we can find the volume:
V = (5.26 × 10^(-3) kg) / (636.0 kg/m^3)
Now we can calculate the buoyant force:
F_b = (density of fluid) × (gravity) × (volume of the immersed object)
= (1000 kg/m^3) × (9.8 m/s^2) × (V)
Next, consider the forces acting on the block of wood when it is in static equilibrium.
The weight of the block (F_g) can be found using the formula: weight = mass × gravity
F_g = (mass of the block) × (gravity)
= (5.26 × 10^(-3) kg) × (9.8 m/s^2)
Since the block is in equilibrium, the buoyant force (F_b) must be equal to the weight (F_g) of the block:
F_b = F_g
Now let's find the stretch of the spring using Hooke's Law:
In equilibrium, the force exerted by the spring (F_s) is equal and opposite to the buoyant force (F_b):
F_s = F_b
Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position:
F_s = -kx
where k is the spring constant and x is the displacement or stretch of the spring.
Setting F_s equal to F_b, we have:
-kx = F_b
Now we can solve for x, the displacement of the spring:
x = -F_b / k
Plug in the values we obtained earlier and calculate:
x = -(F_b) / k
= -(F_g) / k
= -[(5.26 × 10^(-3) kg) × (9.8 m/s^2)] / (15.2 N/m)