# Solving for Block Position After 3.3J Work on Spring-Block System

• mbrmbrg
In summary, the conversation discusses the necessary force of 89 N to hold a block stationary at x = -2.0 cm, and then the block is slowly moved so that the force does +3.3 J of work on the spring-block system. The equation W=-(k\frac{x_f^2}{2}-k\frac{x_i^2}{2}) is used to determine the block's position, which can be +/- 4.3 cm. The equation works because the work done by the person on the spring results in stored elastic potential energy in the spring, and the net work is equal and opposite between the conservative and non-conservative forces. Alternatively, the total work energy method can also be used to determine
mbrmbrg
In the figure below, we must apply a force of magnitude 89 N to hold the block stationary at x = -2.0 cm. From that position we then slowly move the block so that our force does +3.3 J of work on the spring-block system; the block is then again stationary. What is the block's position? (There are two answers.)

The figure is a block resting on a horizontal surface, attatched to a horizontal spring.

-our force=spring force=-kd
so -89J=-k(-0.02m)==>k=-4450N/m

I then said that $$W=-(k\frac{x_f^2}{2}-k\frac{x_i^2}{2})$$, subbed in W=3.3 J, k=-4450N/m, x_i=-0.02m, and got the right answer (which is +/- 4.3cm).
But why did that equation work? 3.3 J is our work, not the spring's?

mbrmbrg said:
In the figure below, we must apply a force of magnitude 89 N to hold the block stationary at x = -2.0 cm. From that position we then slowly move the block so that our force does +3.3 J of work on the spring-block system; the block is then again stationary. What is the block's position? (There are two answers.)

The figure is a block resting on a horizontal surface, attatched to a horizontal spring.

-our force=spring force=-kd
so -89J=-k(-0.02m)==>k=-4450N/m

I then said that $$W=-(k\frac{x_f^2}{2}-k\frac{x_i^2}{2})$$, subbed in W=3.3 J, k=-4450N/m, x_i=-0.02m, and got the right answer (which is +/- 4.3cm).
But why did that equation work? 3.3 J is our work, not the spring's?
Yes, that is the work done by the person on the spring, The work results in stored elastic potential energy in the spring. If you look at the work energy theorem
$$W_{net} = \Delta K.E.$$ and since there is no KE change,
$$W_{net} = 0$$. But the net work is the sum of the work by the conservative forces (spring force) and non conservative forces (the 'we' pushing force). They must therefore be equal and opposite. We do so many joules of work, and the spring force does the same in the other direction.
Alternatively, you can use the total work energy method
$$KE_i + PE_{gravity}_i +PE_{spring}_i = KE_f + PE_{gravity}_f +PE_{spring}_f + W_{n.c.}$$
and since there is no kinetic or gravitational potential energy change, you get the same result for the work done by "we".

The equation you used, W=-(k\frac{x_f^2}{2}-k\frac{x_i^2}{2}), is known as the work-energy theorem. It states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the external force (89 N) is equal to the change in the potential energy of the spring-block system. This change in potential energy is due to the displacement of the block from its initial position (-0.02m) to its final position (x_f). By setting the work done by the external force equal to the change in potential energy, you can solve for the final position of the block. This equation works because energy is conserved in a closed system, and in this case, the only external force acting on the system is the external force applied to the block. Therefore, the work done by this external force is equal to the change in potential energy of the system.

## 1. How do you solve for the block position in a spring-block system after 3.3J of work has been done?

To solve for the block position, you will need to use the equation: x = (W - F_s) / k, where x is the block position, W is the work done (in this case, 3.3J), F_s is the spring force, and k is the spring constant. Plug in the given values and solve for x.

## 2. What is the significance of 3.3J of work in a spring-block system?

3.3J of work represents the amount of energy that has been transferred to the system. This energy is stored in the spring and can be used to calculate the block position.

## 3. How does the spring constant affect the block position in a spring-block system?

The spring constant, denoted by k, determines the stiffness of the spring. A higher spring constant means a stiffer spring, which will result in a smaller block position for a given amount of work done on the system.

## 4. Can the block position be negative in a spring-block system?

Yes, the block position can be negative. This means that the block has moved to the left of its equilibrium position. A positive block position indicates that the block has moved to the right of its equilibrium position.

## 5. How does the mass of the block affect the block position in a spring-block system?

The mass of the block does not directly affect the block position in a spring-block system. However, it does affect the spring force, which is a factor in the equation to solve for the block position. A heavier block will result in a larger spring force and therefore a smaller block position for a given amount of work done on the system.

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