An elastic string stretches to a total length of 30cm under a load of 500N. With an additional load of 100N, the string stretches by a further 2cm. The natural length of the string is??? Solve step by step
F = kx, so
∆F = k∆x
100 = 2k
k = 50 N/cm
So a load of 500N stretched the string 10 cm, making the original length 20 cm.
F = k x , linear stretch versus force
so how much stretch with 500 N?
(500/100) * 2 cm = 10 cm
30 cm - 10 cm = 20 cm
To find the natural length of the string, we can start by determining the amount of stretch caused by the initial load of 500N.
Let's denote the natural length of the string as "L" and the stretch caused by the 500N load as "x."
According to Hooke's Law, the extension of an elastic string is directly proportional to the load applied. So we can write the equation:
x ∝ F
Here, x represents the stretch and F represents the load.
So we can write:
x = kF
Where k is the spring constant.
Since it is mentioned that the string stretches to a total length of 30cm under the load of 500N, we can substitute these values into the equation to find the value of k.
30cm = k * 500N
Now, let's solve for k:
k = 30cm / 500N
k = 0.06cm/N
Now, we can use this spring constant to find the additional stretch caused by the 100N load.
The stretch caused by the additional 100N load is given as 2cm. Let's denote this additional stretch as "y."
y = k * F
y = 0.06cm/N * 100N
y = 6cm
Therefore, the total stretch caused by the additional load of 100N is 6cm.
To find the natural length of the string, we will subtract the additional stretch from the total stretched length:
Natural length of the string = Total stretched length - Additional stretch
Natural length = 30cm - 6cm
Natural length = 24cm
Hence, the natural length of the string is 24cm.
To solve this problem, we'll use Hooke's Law, which states that the extension of an elastic material is directly proportional to the applied force.
Step 1: Determine the spring constant (k)
Hooke's Law can be written as F = k * x, where F is the applied force, k is the spring constant, and x is the extension of the spring.
We have two sets of data:
- Under a load of 500N, the string stretches by 30cm.
- Under an additional load of 100N, the string stretches by 2cm.
Let's use the first set of data to find the spring constant:
500N = k * 30cm
First, let's convert the length to meters:
30cm = 0.30m
Now we can solve for k:
k = 500N / 0.30m
k ≈ 1666.67 N/m
Step 2: Find the natural length of the string
The natural length of the string is the length when no external force is applied. In this case, it will be the length of the string without any load applied.
Let's use the second set of data to find the additional extension caused by the 100N load:
100N = k * 2cm
Again, let's convert the length to meters:
2cm = 0.02m
Solving for the extension caused by the 100N load:
0.02m = 100N / k
0.02m = 100N / 1666.67 N/m
Now we can find the length of the string without any load applied:
Natural length of the string = Total length - 2cm
Natural length of the string = 30cm - 2cm
Natural length of the string ≈ 28cm
So, the natural length of the string is approximately 28cm.