Answer :
A) 5000 N/m
The force constant of the spring can be found by using the expression for the elastic potential energy stored in the spring (which is equal to the work done on it):
[tex]W=U=\frac{1}{2}kx^2[/tex]
where
k is the spring constant
x is the stretching/compression of the spring
In this problem, we have
W = 16.0 J is the work done
x = 8.00 cm = 0.08 m is the stretching
Substituting into the formula and re-arranging it, we find
[tex]k=\frac{2W}{x^2}=\frac{2(16.0 J)}{(0.08 m)^2}=5000 N/m[/tex]
B) 400 N
The magnitude of the force needed to stretch the spring by x = 8.00 cm = 0.08 m is given by Hook's law:
[tex]F=kx[/tex]
where k=5000 N/m as we found previously. Substituting x=0.08 m, we find:
[tex]F=(5000 N/m)(0.08 m)=400 N[/tex]
C) 4 J
The work done to compress the spring by x=4.00 cm=0.04 m is given by the same formula used for part A:
[tex]W=\frac{1}{2}kx^2[/tex]
where in this case, k=5000 N/m and x=0.04 m. Substituting, we find
[tex]W=\frac{1}{2}(5000 N/m)(0.04 m)^2=4 J[/tex]
D) 200 N
As we did in part B), the force needed to stretch this distance is given by Hook's law:
[tex]F=kx[/tex]
where in this case, k=5000 N/m and x=0.04 m. Substituting, we find
[tex]F=(5000 N/m)(0.04 m)=200 N[/tex]
The work and force constant of the spring to stretch it by 8 cm from its unstretched length is,
- A)The force constant of this spring is 5000 N/m.
- B)The magnitude force is needed to stretch the spring 8.00 cm from its unstretched length is 400 N.
- C) The work must be done to compress this spring 4.00 cm from its unstretched length is 4 J.
- D)Force is needed to stretch it this distance is 200 N.
How to calculate work done required to stretch a spring?
Work done on the spring is the product of average force and displacement. It can be given as,
[tex]W=\dfrac{1}{2}kx^2[/tex]
Here, (k) is the spring constant.
The spring constant can be given as,
[tex]k=\dfrac{F}{d}[/tex]
Here, (F) is the force, and (d)is the displacement of the spring.
- A)The force constant of this spring-
The work done to stretch the spring is 16.0 J and the displacement of the spring is 8 cm. Thus, put the values in the above formula as,
[tex]16=\dfrac{1}{2}k(0.08)^2\\k=5000\rm N/m[/tex]
Thus, the force constant of this spring is 5000 N/m.
- B)The magnitude force is needed to stretch the spring 8.00 cm from its unstretched length-
The force constant of this spring is 5000 N/m and displacement of the spring is 8 cm. Thus, put the values in the above formula as,
[tex]5000=\dfrac{F}{0.08}\\F=400\rm N[/tex]
Thus, the magnitude force is needed to stretch the spring 8.00 cm from its unstretched length is 400 N.
- c) The work must be done to compress this spring 4.00 cm from its unstretched length-
The force constant of the spring is 5000 N/m and the displacement of the spring is 4 cm. Thus, put the values in the above formula as,
[tex]W=\dfrac{1}{2}(5000)(0.04)^2\\W=4\rm J[/tex]
Thus, the work must be done to compress this spring 4.00 cm from its unstretched length is 4 J.
- D) Force is needed to stretch it this distance-
The force constant of this spring is 5000 N/m and the displacement of the spring is 4 cm. Thus, put the values in the above formula as,
[tex]5000=\dfrac{F}{0.04}\\F=200\rm N[/tex]
Thus, the magnitude force is needed to stretch the spring 4.00 cm from its unstretched length is 200 N.
The work and force constant of the spring to stretch it by 8 cm from its unstretched length is,
- A)The force constant of this spring is 5000 N/m.
- B)The magnitude force is needed to stretch the spring 8.00 cm from its unstretched length is 400 N.
- C) The work must be done to compress this spring 4.00 cm from its unstretched length is 4 J.
- D)Force is needed to stretch it this distance is 200 N.
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