Hooke S Law And Springs

To find the tension in the spring for the given elongation of , let's use Hooke's Law, which states that the tension or force applied to a spring is directly proportional to its elongation, i.e., , where is the spring constant.

1. Under a tension of 5 N, the elongation is . Therefore, from Hooke's Law, we have:
2. Under a tension of 7 N, the elongation is . Therefore:
3. We need to find the tension for an elongation of . First, let's express this elongation in terms of the known values:
4. Substitute the expressions for and from the above equations:
5. Now substitute these back into the formula for :

Therefore, the tension in the spring for the elongation of is 11 N. The correct answer is 11 N.

To solve this problem, we need to use the conservation of mechanical energy principle as the force involved is conservative and there is no energy loss due to friction.

The mechanical energy in a spring-mass system is the sum of kinetic energy (due to motion) and potential energy (stored in the spring).

The initial configuration of the system is when the spring is compressed to 1 m, and the block is released from rest.

1. Initially, the spring is compressed from its natural length (2 m) to 1 m. So, the initial compression .
2. The potential energy stored in the spring when compressed by is given by: .
3. Initially, the block is at rest, so its kinetic energy .
4. When the block is at a distance from the wall, the spring is stretched by (since 2 m is the natural length).
5. At this point, the potential energy of the spring is: .
6. Let the speed of the block at this point be . Then, its kinetic energy is: .
7. Using conservation of mechanical energy: .

Substituting the known values:

Simplifying for :

Thus, the speed is:

Therefore, the speed of the block at a distance from the wall is , which matches the correct option.