A man steps on his bathroom scale and obtains a reading of 243 lb. The spring in the scale is compressed by a displacement of -0.0480 inches. Calculate the value of its spring constant in (a) pounds per inch (b) newtons per meter.
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.
Hooke's Law equation is given by: F = k * x
where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring.
Given information:
- Force (F) = 243 lb
- Displacement (x) = -0.0480 inches
(a) To find the value of the spring constant in pounds per inch:
We need to rearrange the Hooke's Law equation to solve for the spring constant:
k = F / x
Substituting the values:
k = 243 lb / -0.0480 inches
To simplify the calculation, we will convert inches to feet:
1 inch = 1/12 feet
k = 243 lb / (-0.0480 inches * 1/12 feet)
k = 243 lb / (-0.0480/12 feet)
k = 243 lb / (-0.0040 feet)
k ≈ -60750 lb/ft
Therefore, the spring constant in pounds per inch is approximately -60750 lb/ft.
(b) To find the value of the spring constant in newtons per meter:
We need to convert the force from pounds to newtons (1 lb ≈ 4.44822 N) and the displacement from inches to meters (1 inch ≈ 0.0254 m).
Then we can use the Hooke's Law equation to calculate the spring constant:
k = F / x
Substituting the values:
k = (243 lb * 4.44822 N/lb) / (-0.0480 inches * 0.0254 m/inch)
k = (1079.78966 N) / (-0.0012192 m)
k ≈ -886739 N/m
Therefore, the spring constant in newtons per meter is approximately -886739 N/m.
To calculate the spring constant, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its original position.
Hooke's Law can be written as:
F = -kx
Where:
F is the force applied by the spring
k is the spring constant
x is the displacement of the spring
In this case, the displacement (x) is given as -0.0480 inches, and the force (F) is the weight of the man, which is 243 lb.
(a) To calculate the spring constant in pounds per inch (lb/in), we need to rearrange Hooke's Law equation to solve for k:
k = -F / x
Substituting the given values:
k = -243 lb / -0.0480 in
Simplifying the equation:
k = 243 lb / 0.0480 in
So, the spring constant in pounds per inch is approximately 5062.5 lb/in.
(b) To convert the spring constant to newtons per meter (N/m), we need to convert pounds to Newtons and inches to meters.
1 lb ≈ 4.44822 N (conversion factor)
1 in ≈ 0.0254 m (conversion factor)
Converting the spring constant:
k (N/m) = k (lb/in) * (4.44822 N/lb) / (0.0254 m/in)
Substituting the given spring constant value:
k (N/m) = 5062.5 lb/in * (4.44822 N/lb) / (0.0254 m/in)
Simplifying the equation:
k (N/m) ≈ 113785 N/m
So, the spring constant in newtons per meter is approximately 113785 N/m.