Problem 32 - University Physics with Modern Physics

Question

\(\bullet\) \(\bullet\) To stretch a spring 3.00 \(\mathrm{cm}\) from its unstretched length, 12.0 \(\mathrm{J}\) of work must be done. (a) What is the force constant of this spring? (b) What magnitude force is needed to stretch the spring 3.00 \(\mathrm{cm}\) from its unstretched length? (c) How much work must be done to compress this spring 4.00 \(\mathrm{cm}\) from its unstretched length, and what force is needed to compress it this distance?

Step-by-Step Solution

Answer

(a) 26667 N/m; (b) 800 N; (c) 21.33 J and 1066.67 N.

Step 1: Calculate the Force Constant (Hooke's Law)

The work done on a spring is given by the formula \[ W = \frac{1}{2} k x^2 \]where \( W \) is the work done (12.0 J), \( k \) is the force constant, and \( x \) is the displacement in meters (0.03 m). Rearranging the formula to solve for \( k \), we get:\[ k = \frac{2W}{x^2} = \frac{2 \times 12.0}{(0.03)^2} \] Calculate \( k \).

Step 2: Compute the Force Constant

Substituting the values into the equation:\[ k = \frac{2 \times 12.0}{0.0009} = \frac{24.0}{0.0009} = 26666.67 \ N/m \]The force constant of the spring is approximately 26667 N/m.

Step 3: Calculate Force to Stretch the Spring

The force needed to stretch the spring can be found using Hooke's Law: \[ F = kx \] Substitute the values: \[ F = 26666.67 \times 0.03 \]

Step 4: Compute the Stretch Force

Calculate the force: \[ F = 800 \ N \] A force of 800 N is needed to stretch the spring 3.00 cm.

Step 5: Calculate Work for Compression

The work done to compress the spring 4.00 cm is given by the same formula for work done:\[ W = \frac{1}{2} k x^2 \]where \( x = 0.04 \) m.Substitute the known values to find the work:\[ W = \frac{1}{2} \times 26666.67 \times (0.04)^2 \]

Step 6: Compute Work for Compression

Calculate the work:\[ W = \frac{1}{2} \times 26666.67 \times 0.0016 = 21.33 \ J \]The work done to compress the spring is 21.33 J.

Step 7: Calculate Force for Compression

The force needed to compress the spring 4.00 cm is also found using Hooke's Law:\[ F = kx = 26666.67 \times 0.04 \]

Step 8: Compute Compression Force

Calculate the force:\[ F = 1066.67 \ N \]A force of 1066.67 N is needed to compress the spring 4.00 cm.

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in physics that helps us understand how springs behave when forces are applied to them. This law states that the force needed to extend or compress a spring by a certain distance is directly proportional to that distance.
Imagine a spring stretched or compressed from its natural length. Hooke's Law gives us the equation:

• \( F = kx \)

where \( F \) is the force applied, \( k \) is the spring constant (force constant), and \( x \) is the change in length from its natural position. The spring constant \( k \) measures the stiffness of the spring; a larger \( k \) means a stiffer spring that requires more force to stretch or compress by a certain amount.
When we calculated the force needed to stretch the spring by 3 cm in the exercise, we used Hooke's Law to find that \( k = 26666.67 \text{ N/m} \), which is a high value indicating a very stiff spring.

Work-Energy Principle

The Work-Energy Principle states that the work done on an object results in a change in its energy. For springs, the work done by an external force to stretch or compress the spring is stored as potential energy in the spring.
The formula used to calculate the work done on a spring is:

• \( W = \frac{1}{2} k x^2 \)

This equation tells us that the work done is equal to half the product of the spring constant \( k \) and the square of the displacement \( x \).
For the exercise problem: stretching the spring by 3 cm required work of 12 J, and for compressing it by 4 cm, we computed that 21.33 J of work was needed. The potential energy stored in the spring will be released when the spring returns to its natural state.

Force Calculation

Force calculation in spring scenarios is essential to understanding how much effort is required to stretch or squeeze a spring. This is again where Hooke's Law is very useful. To compute the force needed, we multiply the spring constant \( k \) by the displacement \( x \):

• \( F = kx \)

In this exercise, when the spring is stretched by 3 cm, applying Hooke's Law gives us a required force of \( 800 \text{ N} \).
Similarly, when calculating the force necessary to compress the spring by 4 cm, an application of Hooke's Law resulted in a force of \( 1066.67 \text{ N} \).
Both these calculations reveal how altering the displacement changes the force, directly reflecting the spring's resistance to deformation.