A spring has k = 88 N/m. Use a graph to determine the work needed to stretch it from x = 3.6 cm to x = 5.6 cm, where x is the displacement from its unstretched length.
It's because, you the equation is
1/2(K)(x^2)
So you change the cm to meters, and then you subtract, getting (.02) then you get (.02)^2, cuz u have to square it,
Then you * by K which is 88, and then divide by 2. That gives you the SPE, (spring potential energy)
Oh, spring, tell me about your stretchy adventures! Now, let's see what kind of work we need to do to stretch you from 3.6 cm to 5.6 cm.
Well, on a graph, the displacement is usually plotted on the x-axis, and the force applied on the y-axis. But since you're a spring, we have displacement on the x-axis and the work done on the y-axis.
To calculate the work done, we'll need to know the area under the curve on the graph. Since the graph is a straight line, we can use the equation W = (1/2)kx^2 to find the work done.
First, let's find the displacement change: 5.6 cm - 3.6 cm = 2 cm.
Now, we plug that into the equation:
W = (1/2)(88 N/m)(0.02 m)^2
Let's do some math:
W = 0.5 × 88 N/m × (0.02 m)^2
= 0.5 × 88 N/m × 0.0004 m^2
= 0.5 × 0.0352 N.m
= 0.0176 N.m
So, according to my calculations, the work needed to stretch the spring from 3.6 cm to 5.6 cm is approximately 0.0176 N.m. Keep stretching, my springy friend!
To determine the work needed to stretch a spring from one point to another, we need to calculate the area under the force-displacement graph. In this case, since the spring constant (k) is given, we can use Hooke's Law to obtain the force as a function of displacement.
Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. Mathematically, it can be expressed as:
F = - kx
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, the spring constant (k) is given as 88 N/m, and we need to determine the work done in stretching the spring from x = 3.6 cm to x = 5.6 cm.
Step 1: Convert the displacements from centimeters to meters.
3.6 cm = 0.036 m
5.6 cm = 0.056 m
Step 2: Calculate the force at each displacement using Hooke's Law.
F1 = -k * x1 = -88 N/m * 0.036 m
F2 = -k * x2 = -88 N/m * 0.056 m
Step 3: Plot a graph of force (y-axis) against displacement (x-axis) with the two given points.
|
F2|____________________________________
| /
| /
F1|______________________________/
| x1 x2
Step 4: Calculate the area under the force-displacement graph. Since the graph is a triangle, we can use the formula for the area of a triangle:
Area = 1/2 * base * height
In this case, the base is the displacement (x2 - x1) and the height is the average of the two forces (F1 + F2) / 2.
Area = 1/2 * (x2 - x1) * ((F1 + F2) / 2)
= 1/2 * (0.056 m - 0.036 m) * ((-88 N/m * 0.036 m + -88 N/m * 0.056 m) / 2)
Now, plug in the values:
Area = 1/2 * 0.02 m * ((-88 N/m * 0.036 m + -88 N/m * 0.056 m) / 2)
Calculating the expression inside the parentheses:
(-88 N/m * 0.036 m + -88 N/m * 0.056 m) / 2 = -0.088 N
Now, substitute back into the equation:
Area = 1/2 * 0.02 m * (-0.088 N)
= -0.0022 Nm
The work needed to stretch the spring from x = 3.6 cm to x = 5.6 cm is approximately -0.0022 Nm (negative sign indicates that work is done against the force of the spring).
but when you did this.... k[(.056)^2 - (.036)^2]
k = 88 right?
88(.056^2 - .036^2)
then the answer would be .16192? it said that its incorrect. im alittle confused?
You don't really need to draw a graph to figure that out. We can't draw them for you here, anyway. You will have to convert the 3.6 cm to 0.036 m and the 5.6 cm to 0.056 m.
The answer is k[(.056)^2 - (.036)^2]
If you used a graph to do it, you should plot the force kx vs x and measure tha area under the curve from x = 0.036 to x = 0.056. You should get the same answer, since the area is a trapezoid with width 0.020 and average height 0.046.