An elastic cord is 65 cm long when a weight of 75 N hangs from it but is 85 cm long when a weight of 189 N hangs from it. What is the “spring” constant k of this cord
k=changeinForce/change in length.
k=(189-75)/(85-65) N/cm
To find the spring constant (k) of the cord, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to its extension or compression.
Hooke's Law can be expressed as:
F = k * x
Where:
F is the force applied to the spring
k is the spring constant
x is the extension or compression of the spring
In this case, we have two different situations:
Situation 1:
Extension (x1) = 65 cm
Force (F1) = 75 N
Situation 2:
Extension (x2) = 85 cm
Force (F2) = 189 N
Using Hooke's Law, we can set up two equations:
F1 = k * x1
F2 = k * x2
Rearranging the equations, we get:
k = F1 / x1
k = F2 / x2
Substituting the given values, we have:
k = 75 N / 65 cm
k = 189 N / 85 cm
To calculate the spring constant (k), convert the given extensions from centimeters to meters:
k = 75 N / (0.65 m)
k = 189 N / (0.85 m)
Evaluating these expressions, we find:
k ≈ 115.38 N/m
k ≈ 222.35 N/m
Thus, the "spring" constant (k) of this cord is approximately 115.38 N/m and 222.35 N/m.
To find the spring constant (k) of the elastic cord, we can use Hooke's Law, which states that the force (F) exerted on a spring is directly proportional to the displacement (x) from its equilibrium position, and the constant of proportionality is the spring constant (k). Mathematically, it can be expressed as:
F = -kx
First, we need to determine the displacement (x) for each situation. The displacement is the change in length of the cord from its original length (also known as the extension).
For the first situation:
Original length (L₁) = 65 cm
Weight (F₁) = 75 N
Since we know that the weight is directly proportional to the extension, we can calculate the extension (x₁) using the formula:
F₁ = kx₁
x₁ = F₁ / k
For the second situation:
Original length (L₂) = 85 cm
Weight (F₂) = 189 N
Using the same formula, we can calculate the extension (x₂):
x₂ = F₂ / k
Now, equating the extensions for both situations, we have:
x₁ = x₂
F₁ / k = F₂ / k
Since the spring constant is the same, we can rearrange the equation as follows:
F₁ / x₁ = F₂ / x₂
Plugging in the values we have:
75 N / x₁ = 189 N / x₂
Now, we can solve for x₁ and x₂ by setting up a proportion:
x₁ / 75 N = x₂ / 189 N
Cross-multiplying gives us:
x₁ * 189 N = 75 N * x₂
x₁ * 189 N = 75 N * x₂
Simplifying the equation:
189 x₁ = 75 x₂
Now, substitute the known values for x₁ and x₂:
189 * (65 cm - L₁) = 75 * (85 cm - L₂)
Now, we can solve for k:
k = F₁ / x₁
Substitute the values for F₁ and x₁:
k = 75 N / (65 cm - L₁)
After calculating L₁ and L₂, plug in the values to find k.
Note: Make sure to convert the lengths to meters (m) if you want the spring constant in SI units.