In a room that is 2.78 m high, a spring (unstrained length = 0.30 m) hangs from the ceiling. A board whose length is 1.94 m is attached to the free end of the spring. The board hangs straight down, so that its 1.94-m length is perpendicular to the floor. The weight of the board (104 N) stretches the spring so that the lower end of the board just extends to, but does not touch, the floor. What is the spring constant of the spring?
To find the spring constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. The equation for Hooke's Law can be written as follows:
F = -kx
where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, the weight of the board stretches the spring. When the lower end of the board just extends to the floor, the displacement from the equilibrium position is the length of the spring (0.30 m).
So, we have:
104 N = -k * 0.30 m
To solve for the spring constant (k), we can rearrange the equation:
k = -104 N / 0.30 m
k = -346.67 N/m
The negative sign indicates that the force exerted by the spring is in the opposite direction to its displacement. However, for the sake of simplicity, we can consider the magnitude of the spring constant: 346.67 N/m.
Therefore, the spring constant of the spring is approximately 346.67 N/m.