A 0.63-kg metal sphere oscillates at the end of a vertical spring. As the spring stretches from 0.13 to 0.22 m (relative to its unstrained length), the speed of the sphere decreases from 5.95 to 4.75 m/s. What is the spring constant of the spring? (The sphere hangs from the bottom of the spring.)
Answer in N/m
To find the spring constant of the spring, we can use the concept of conservation of mechanical energy.
The mechanical energy of the system is given by the sum of kinetic energy and potential energy:
E = KE + PE
Since the sphere oscillates, we know that the mechanical energy is constant throughout the motion. Thus, we can set the initial mechanical energy equal to the final mechanical energy:
E_initial = E_final
Initially, the sphere has a velocity of 5.95 m/s, which corresponds to kinetic energy:
KE_initial = (1/2) * m * v_initial^2
where m is the mass of the sphere (0.63 kg) and v_initial is the initial velocity (5.95 m/s).
Finally, the sphere is at its maximum displacement when the spring is stretched by 0.22 m, so we can calculate the potential energy due to the spring's elongation:
PE_max = (1/2) * k * x_max^2
where k is the spring constant (in N/m) and x_max is the maximum displacement (0.22 m).
Similarly, at the minimum displacement (0.13 m), the potential energy due to the spring's elongation can be calculated:
PE_min = (1/2) * k * x_min^2
where x_min is the minimum displacement (0.13 m).
Now, we can write the equation for the conservation of mechanical energy:
KE_initial + PE_max = KE_final + PE_min
Substituting the values, we have:
(1/2) * m * v_initial^2 + (1/2) * k * x_max^2 = (1/2) * m * v_final^2 + (1/2) * k * x_min^2
Plugging in the values given in the problem, we have:
(1/2) * (0.63 kg) * (5.95 m/s)^2 + (1/2) * k * (0.22 m)^2 = (1/2) * (0.63 kg) * (4.75 m/s)^2 + (1/2) * k * (0.13 m)^2
Now we can solve this equation to find the value of k, the spring constant.
To find the spring constant of the spring, we can use the formula for the potential energy stored in a spring when it is stretched or compressed:
Potential energy = (1/2) * k * x^2
Where:
k is the spring constant
x is the displacement from the equilibrium position
The speed of the sphere decreasing tells us that the kinetic energy is being converted into potential energy, while the change in spring length tells us the change in displacement.
Given that the mass of the sphere is 0.63 kg, the change in speed is 5.95 m/s - 4.75 m/s = 1.2 m/s, and the change in displacement is (0.22 m - 0.13 m) = 0.09 m, we can set up the equation using the principle of conservation of energy:
(1/2) * m * (v_final^2 - v_initial^2) = (1/2) * k * (x_final^2 - x_initial^2)
Plugging in the values:
(1/2) * (0.63 kg) * (1.2 m/s)^2 = (1/2) * k * (0.09 m)^2
Simplifying the equation:
0.378 J = (1/2) * k * 0.0081 m^2
Now, we can solve for k:
k = (0.378 J) / (0.0081 m^2)
Calculating this value gives:
k ≈ 46.67 N/m
Therefore, the spring constant of the spring is approximately 46.67 N/m.
The increase in potential energy in stretching from 0.13 to 0.22 m equals the decrease in kinetic energy of the metal sphere. That will let you solve for the spring constant k.
k [(.22)^2 -(.13)^2] = (m/2)[(5.95)^2 - (4.75)^2]
You know the mass m, so solve for k.,