A person who weighs 829 N steps onto a spring scale in the bathroom, and the spring compresses by 0.538 cm. (a) What is the spring constant? (b) What is the weight of another person who compresses the spring by 0.390 cm?
(a)W=kx
k=W/x=829/0.538 = 1540.9 N/m
(b)W₁=kx₁=1540.9•0.390 =600 N
Adjust for the fact that the compression is .538cm, not .538m
To find the spring constant, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, Hooke's Law is expressed as:
F = -kx
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring.
(a) To find the spring constant, we need to rearrange Hooke's Law equation:
k = -F / x
We know that the weight of the person is 829 N, and the displacement of the spring is 0.538 cm (which needs to be converted to meters):
x = 0.538 cm = 0.538 / 100 = 0.00538 m
Substituting the values into the equation, we get:
k = -829 N / 0.00538 m
Calculating this, we find:
k ≈ -154,079 N/m (since the negative sign indicates the restoring force of the spring)
So, the spring constant is approximately 154,079 N/m.
(b) Now, to find the weight of another person who compresses the spring by 0.390 cm, we can use the same formula:
F = -kx
We need to find the value of F when x = 0.390 cm:
x = 0.390 cm = 0.390 / 100 = 0.00390 m
Substituting the values into the equation, we get:
F = -154,079 N/m * 0.00390 m
Calculating this, we find:
F ≈ -601.049 N
Since the negative sign in the equation indicates the restoring force of the spring, we ignore it and take the magnitude of the force:
|F| ≈ 601.049 N
Therefore, the weight of the other person who compresses the spring by 0.390 cm is approximately 601 N.