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.)
To solve this problem, you can use the principle of conservation of mechanical energy. Let's break down the steps to find the spring constant.
Step 1: Calculate the potential energy at the two extremes.
The potential energy of the system is given by the formula: U = (1/2)kx^2, where U is the potential energy, k is the spring constant, and x is the displacement.
At the first extreme, when the spring is stretched to 0.10 m:
U1 = (1/2)k(0.10)^2 = 0.005k
At the second extreme, when the spring is stretched to 0.21 m:
U2 = (1/2)k(0.21)^2 = 0.02205k
Step 2: Calculate the kinetic energy at the two extremes.
The kinetic energy of the system is given by the formula: K = (1/2)mv^2, where K is the kinetic energy, m is the mass of the sphere, and v is the speed.
At the first extreme, when the sphere has a speed of 5.35 m/s:
K1 = (1/2)(0.67)(5.35)^2 = 9.024795 J
At the second extreme, when the sphere has a speed of 4.55 m/s:
K2 = (1/2)(0.67)(4.55)^2 = 7.006855 J
Step 3: Apply the principle of conservation of mechanical energy.
According to the principle of conservation of mechanical energy, the sum of the potential energy and kinetic energy of the system remains constant.
Therefore, at the first extreme:
U1 + K1 = Constant
0.005k + 9.024795 = C1 ----(1)
Similarly, at the second extreme:
U2 + K2 = Constant
0.02205k + 7.006855 = C2 ----(2)
Step 4: Solve for the spring constant.
Since the constants C1 and C2 should be the same, we can equate equations (1) and (2) to solve for k.
0.005k + 9.024795 = 0.02205k + 7.006855
Simplifying the equation, we get:
0.01705k = 2.01794
Dividing both sides by 0.01705, we find:
k = 118.50 N/m
Therefore, the spring constant of the spring is approximately 118.50 N/m.
To find the spring constant of the spring, we can use the formula for the potential energy of a spring, which is given by:
Potential energy (U) = (1/2) * k * x^2
Where:
- U is the potential energy
- k is the spring constant
- x is the displacement from the equilibrium position
The potential energy of the spring is equal to the kinetic energy of the sphere, given by:
Kinetic energy (K) = (1/2) * m * v^2
Where:
- K is the kinetic energy
- m is the mass of the sphere
- v is the velocity of the sphere
Since energy is conserved in simple harmonic motion, the initial potential energy of the spring is equal to the final kinetic energy of the sphere. Therefore, we can set the two equations equal to each other:
(1/2) * k * x^2 = (1/2) * m * v^2
Plugging in the given values:
- m = 0.67 kg
- x1 = 0.10 m
- x2 = 0.21 m
- v1 = 5.35 m/s
- v2 = 4.55 m/s
We can rewrite the equation as:
k * x1^2 = m * v2^2 - m * v1^2
Substituting the values:
k * 0.10^2 = 0.67 * (4.55^2 - 5.35^2)
Simplifying:
k * 0.01 = 0.67 * (-4.56 + 28.56)
k * 0.01 = 0.67 * 24
k * 0.01 = 16.08
Dividing both sides by 0.01:
k = 1608 N/m
Therefore, the spring constant of the spring is 1608 N/m.