A spiral spring of natural length 25cm has a scale pan hanging freely in its lower end. When an object of mass 45g is placed in the pan ,its length becomes 26.80cm. When another object of mass 65g is placed in the pan, the length becomes 27.05cm. Calculate the mass of the scale pan.?
F = k x
(45+m) = k (26.8-25) = 1.8 k
(65+m) = k(27.05 -25) = 2.05 k
(2.05 /1.8)[ (45+m) ] = 2.05 k
is
51.25 + 1.14 m = 2.05 k
65 + m = 2.05 k
--------------------------- sutract
- 13.75 + .14 m = 0
m = 98.2 grams
check my arithmetic !
To solve this problem, we need to use Hooke's law, which states that the extension of a spring is directly proportional to the force applied to it. In this case, the force is the weight of the objects placed in the scale pan.
Let's start by finding the spring constant, k, which is a measure of how stiff the spring is. The formula is:
k = (F / x)
Where:
- k is the spring constant
- F is the force applied to the spring
- x is the extension of the spring
For the first object:
F1 = mass1 × g
x1 = length1 - natural_length
Where:
- mass1 is the mass of the first object (45g)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- length1 is the extended length when the first object is placed in the pan
For the second object:
F2 = mass2 × g
x2 = length2 - natural_length
Where:
- mass2 is the mass of the second object (65g)
- length2 is the extended length when the second object is placed in the pan
Now let's calculate the spring constant and find the mass of the scale pan:
1. Calculate the spring constant:
k = F1 / x1
k = (mass1 × g) / (length1 - natural_length)
k = F2 / x2
k = (mass2 × g) / (length2 - natural_length)
2. Use the spring constant to find the mass of the scale pan:
Let mass_scalepan be the mass of the scale pan.
For the scale pan, when no objects are placed in it, the extension is zero:
(0) = (mass_scalepan × g) / (0 - natural_length)
Solving for mass_scalepan:
0 = -(mass_scalepan × g) / natural_length
mass_scalepan × g = 0
Therefore, the mass of the scale pan should be 0 grams, meaning it has negligible mass compared to the objects being weighed.