3. An elastic string hangs from a nail N. When as mass of m grams is attached to its lower end, the elastic is stretched so that its lower length is x cm, as shown in the diagram:
N The table shows result of the two experiments:
Length (x cm) 43 49
Mass (m grams) 50 80
m
It is known that x and m are connected by the equation x = c + dm, where c and d are constants.
(a) Solve your equations to find the value of c and the value of d.
(b) Find the mass at the end of the string when its length is 40 cm.
(c) What does the value of c represents?
a. c=-33, d=0.2
b.46.51
c. Extension
Not able to solve
a) To find the values of c and d, we can use the given data points to create a system of equations:
For experiment 1:
x = c + dm
43 = c + d(50)
For experiment 2:
x = c + dm
49 = c + d(80)
Let's solve this system of equations using the method of substitution:
From the first equation, we can express c in terms of d:
c = 43 - 50d
Substituting this expression for c in the second equation:
49 = (43 - 50d) + d(80)
49 = 43 - 50d + 80d
49 - 43 = 30d
6 = 30d
d = 6/30
d = 1/5
Now, substituting this value of d back into the first equation:
43 = c + (1/5)(50)
43 = c + 10
c = 43 - 10
c = 33
Therefore, the value of c is 33 and the value of d is 1/5.
b) To find the mass when the length is 40 cm, we can use the equation x = c + dm:
40 = 33 + (1/5)m
7 = (1/5)m
m = 7 * 5
m = 35 grams
Therefore, the mass at the end of the string when its length is 40 cm is 35 grams.
c) The value of c represents the length of the string when there is no mass attached to it. It is the constant term in the equation x = c + dm. In this case, c = 33, so when there is no mass attached, the length of the string is 33 cm.
To find the values of c and d, we can use the given information from the table.
(a) From the equation x = c + dm, we can set up two equations using the values from the table:
For the first experiment (x = 43 cm, m = 50 g):
43 = c + d * 50
For the second experiment (x = 49 cm, m = 80 g):
49 = c + d * 80
We now have a system of two equations with two variables (c and d). We can solve this system of equations to find the values of c and d.
Subtracting the first equation from the second equation, we get:
49 - 43 = c + d * 80 - (c + d * 50)
6 = d * 80 - d * 50
6 = d * (80 - 50)
6 = d * 30
d = 6 / 30
d = 0.2
Substituting the value of d into the first equation:
43 = c + 0.2 * 50
43 = c + 10
c = 43 - 10
c = 33
Therefore, the value of c is 33 and the value of d is 0.2.
(b) To find the mass at the end of the string when its length is 40 cm, we can use the equation x = c + dm and substitute the values of c, d, and x:
40 = 33 + 0.2 * m
Simplifying the equation:
40 - 33 = 0.2 * m
7 = 0.2 * m
Dividing both sides by 0.2:
7 / 0.2 = m
m = 35
So, the mass at the end of the string when its length is 40 cm is 35 grams.
(c) The value of c represents the unstretched length of the elastic string. It is the length of the string when there is no mass attached to it. In this case, the value of c is 33 cm.
To solve the equations and find the values of c and d, we can use the given information from the experiments.
(a) Since we have two sets of values for x and m from the experiments, we can substitute them into the equation x = c + dm and form two equations:
43 = c + 50d (1)
49 = c + 80d (2)
To solve this system of equations, we can use the method of substitution or elimination.
Using substitution method:
From equation (1), we can express c in terms of d: c = 43 - 50d
Substituting this into equation (2):
49 = (43 - 50d) + 80d
49 = 43 + 80d - 50d
49 = 43 + 30d
6 = 30d
d = 6/30
d = 0.2
Now, substituting the value of d back into equation (1):
43 = c + 50 * 0.2
43 = c + 10
c = 43 - 10
c = 33
So, the value of c is 33 and the value of d is 0.2.
(b) To find the mass at the end of the string when its length is 40 cm (let's call it m'), we can use the equation x = c + dm and substitute the given values:
40 = 33 + 0.2m'
40 - 33 = 0.2m'
7 = 0.2m'
m' = 7/0.2
m' = 35
Therefore, the mass at the end of the string when its length is 40 cm is 35 grams.
(c) The value of c represents the initial length of the string, where there is no additional mass attached to it. In this case, when m = 0, we can substitute m = 0 into the equation x = c + dm:
x = c + d * 0
x = c
Thus, c represents the constant hanging length of the elastic string (in centimeters) when there is no additional mass attached to it.