Use Hooke's law equation to answer the question
F_sp =-kx
WHat happens to the displacement vector x when the spring constant k has a higher value and the applied force F_sp remains constant
it remains the same
it changes its direction
it decreases in magnitude
it increases in magitude
The displacement vector x refers to the change in position or elongation of the spring from its equilibrium position. According to Hooke's law equation, for a constant applied force (F_sp), if the spring constant (k) has a higher value, the displacement vector x will decrease in magnitude.
This means that when the spring constant is increased, the spring becomes stiffer and requires a smaller displacement to counteract the same applied force. Therefore, the displacement vector x will be smaller compared to when the spring constant is lower.
When the spring constant k has a higher value and the applied force F_sp remains constant, the displacement vector x decreases in magnitude. This means that the spring compresses or stretches less for the same applied force.
To answer this question using Hooke's Law equation, let's consider the relationship between the displacement vector (x), the spring constant (k), and the applied force (F_sp):
F_sp = -kx
In this equation, the negative sign indicates that the direction of the force exerted by the spring is opposite to the displacement direction (since displacement is a vector).
Now, if we keep the applied force (F_sp) constant and increase the spring constant (k), we can analyze how it affects the displacement vector (x):
1. It remains the same: If the spring constant (k) changes while the applied force (F_sp) remains constant, there won't be any change in the displacement vector (x) unless other factors are involved. So, this option is not likely.
2. It changes its direction: Since displacement is a vector quantity, changing the spring constant (k) with a constant applied force (F_sp) wouldn't affect the direction of the displacement vector (x). The direction depends on the external force applied and the initial position, not the spring constant.
3. It decreases in magnitude: Increasing the spring constant (k) while maintaining a constant applied force (F_sp) will result in a decrease in the magnitude of the displacement vector (x). This is because a higher spring constant implies a stiffer spring, which will resist more against any given force, leading to less displacement.
4. It increases in magnitude: This is the incorrect option. Since we are keeping the applied force (F_sp) constant, increasing the spring constant (k) will not cause the displacement vector (x) to increase in magnitude.
Therefore, the correct answer is: "it decreases in magnitude."