A 0.67-kg metal sphere oscillates at the end of a vertical spring. As the spring stretches from 0.10 to 0.21 m (relative to its unstrained length), the speed of the sphere decreases from 5.35 to 4.55 m/s. What is the spring constant of the spring? (The sphere hangs from the bottom of the spring.)
N/m
K(Ho)^2+(m)(Vo^2)=K(Hf)^2+m(Vo^2)
To find the spring constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
First, let's calculate the change in potential energy and kinetic energy of the sphere as the spring stretches.
The change in potential energy (ΔPE) can be calculated using the formula ΔPE = mgΔh, where m is the mass of the sphere (0.67 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and Δh is the change in height.
Δh = 0.21 m - 0.10 m = 0.11 m
ΔPE = (0.67 kg) * (9.8 m/s²) * (0.11 m) = 0.78619 J
The change in kinetic energy (ΔKE) can be calculated using the formula ΔKE = (1/2) mv₂² - (1/2) mv₁², where m is the mass of the sphere, v₂ is the final velocity, and v₁ is the initial velocity.
ΔKE = (1/2) (0.67 kg) (4.55 m/s)² - (1/2) (0.67 kg) (5.35 m/s)²
= 0.5 * 0.67 kg * 20.6805 m²/s² - 0.5 * 0.67 kg * 28.6225 m²/s²
= 1.60578 J - 2.11222 J
= -0.50644 J
Notice that the change in kinetic energy is negative, indicating that the sphere loses kinetic energy as it slows down.
Now, let's consider the work done by the spring (W_spring) on the sphere.
The work done by the spring is equal to the negative of the change in potential energy, as energy is conserved.
W_spring = -ΔPE
= -0.78619 J
Hooke's Law states that the work done by a spring is given by W_spring = (1/2) kx², where k is the spring constant and x is the displacement of the spring from its equilibrium position.
Since the sphere oscillates from 0.10 to 0.21 m, the displacement x is equal to 0.21 m. Substituting the known values into the equation, we can solve for k.
-0.78619 J = (1/2) k (0.21 m)²
Now, solve for k.
k = (-0.78619 J) / [(1/2) (0.21 m)²]
= (-0.78619 J) / (0.5 * 0.0441 m²)
= -0.78619 J / 0.02205 m²
= -35.7 N/m
Since the spring constant cannot be negative, we take the magnitude of the result, giving us the spring constant:
k = 35.7 N/m
Therefore, the spring constant of the spring is 35.7 N/m.