A modern sculpture has a large horizontal spring, with a spring constant of 277 N/m, that is attached to a 58.0-kg piece of uniform metal at its end and holds the metal at an angle of θ = 48.0° above the horizontal direction. The other end of the metal is wedged into a corner as shown. By how much has the spring stretched?
To find out how much the spring has stretched, we can use Hooke's Law, which states that the force exerted by a spring is proportional to its displacement. The formula for Hooke's Law is:
F = -kx
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring.
In this case, the weight of the metal piece is acting downward and is balanced by the force exerted by the spring which is acting upward at an angle of 48.0° above the horizontal direction. Let's break this force into its vertical and horizontal components.
Vertical component: Fv = mg
where m is the mass of the metal piece and g is the acceleration due to gravity.
Horizontal component: Fh = F * sin(48.0°)
Since the vertical component of the force is balanced by the weight, we have:
mg = Fv
And, since the horizontal component of the force is balanced by the spring force, we have:
Fh = -kx
Now, let's plug in the values given in the problem:
m = 58.0 kg
k = 277 N/m
θ = 48.0°
We know that sin(θ) = Fv/F, so we can substitute this expression for Fv in the equation Fh = -kx:
F * sin(θ) = -kx
Since we are interested in finding the displacement of the spring (x), we can rearrange the equation to solve for x:
x = -(F * sin(θ))/k
Now, let's substitute the known values into the equation to find x:
x = -(mg * sin(θ))/k
x = -((58.0 kg * 9.8 m/s^2) * sin(48.0°))/(277 N/m)
Calculating this expression, we find:
x ≈ -0.79 m
So, the spring has stretched approximately 0.79 meters.
To determine how much the spring has stretched, we need to calculate the displacement of the metal piece from its equilibrium position.
1. Start by drawing a free-body diagram of the metal piece. In this case, we have the weight acting vertically downward, the tension in the spring acting horizontally, and the normal force acting perpendicular to the surface it's wedged into.
2. To calculate the displacement, we need to resolve the weight vector into components. The component of the weight that opposes the tension in the spring is given by mg sin(θ), where m is the mass of the metal piece and θ is the angle above the horizontal direction.
3. The spring force F_s is given by Hooke's Law, which states that the force exerted by a spring is proportional to its displacement. Hooke's Law is expressed as F_s = -kx, where k is the spring constant and x is the displacement.
4. Equating the spring force to the weight component (F_s = mg sin(θ)), we get -kx = mg sin(θ).
5. Rearranging the equation, we have x = -(mg sin(θ))/k.
6. Plug in the values given: m = 58.0 kg, g = 9.8 m/s^2, θ = 48.0°, and k = 277 N/m. Convert the angle to radians (θ = 48.0° * π/180) for trigonometric calculations.
7. Calculate x using the formula: x = -((58.0 kg)(9.8 m/s^2) sin(48.0° * π/180)) / (277 N/m).
8. Solve the equation to find x (the negative sign signifies that the displacement is in the opposite direction of the weight component): x ≈ -0.548 m.
Therefore, the spring has stretched approximately 0.548 meters.