# Spring force equation

The spring force equation, derived from Hooke’s law, is expressed as Fs = k Δx, where Fs represents the spring force, k is the spring constant, and Δx denotes the displacement from the equilibrium position of the spring. This equation enables the calculation of the force exerted by a spring on an object. The sign of Δx indicates whether the spring is stretched (+ve) or compressed (-ve).

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## Practice problems

### Problem #1

Calculate the force exerted by a spring on a block when the block, attached to a spring with a natural length of 25 cm, moves rightwards and stretches the spring by 50 cm. The spring constant is 75 N/m.

Solution

Given data:

• Force exerted by spring on a block, Fs = ?
• Initial length of a spring, x0 = 25 cm = 0.25 m
• Final length of a spring, x = 50 cm = 0.5 m
• Spring constant, k = 75 N/m

Applying the formula:

• Fs = k Δx = k × (x – x0)
• Fs = 75 × (0.5 – 0.25)
• Fs = 75 × 0.25
• Fs = 18.75 N

Therefore, the force exerted by spring on a block is -18.75 N. (left)

### Problem #2

Determine the force exerted by a spring on a block when the block, attached to a spring with a natural length of 15 cm, moves leftwards and compresses the spring by 5 cm. The spring constant is 55 N/m.

Solution

Given data:

• Force exerted by spring on a block, Fs = ?
• Initial length of a spring, x0 = 15 cm = 0.15 m
• Final length of a spring, x = 5 cm = 0.05 m
• Spring constant, k = 55 N/m

Applying the formula:

• Fs = k Δx = k × (x – x0)
• Fs = 55 × (0.05 – 0.15)
• Fs = 55 × (-0.01)
• Fs = -5.5 N

Therefore, the force exerted by spring on a block is 5.5 N. (right)

### Problem #3

A spring is in equilibrium, with its left end fixed to a wall and its right end fixed to a wooden crate. When the wooden crate is moved to the right, the spring stretches from a length of 0.5 m to 0.65 m. Calculate the force exerted by the spring on the wooden crate, given that the spring constant is 150 N/m.

Solution

Given data:

• Initial length of a spring, x0 = 0.5 m
• Final length of a spring, x = 0.65 m
• Force exerted by spring on a wooden crate, Fs = ?
• Spring constant, k = 150 N/m

Applying the formula:

• Fs = k Δx = k × (x – x0)
• Fs = 150 × (0.65 – 0.5)
• Fs = 150 × (0.15)
• Fs = 22.5 N

Therefore, the force exerted by spring on a wooden crate is -22.5 N. (left)

### Problem #4

A spring is in equilibrium, with its left end fixed to a wall and its right end fixed to a wooden crate. When the wooden crate is moved to the left, the spring compresses from a length of 0.5 m to 0.2 m. Calculate the force exerted by the spring on the wooden crate, given that the spring constant is 170 N/m.

Solution

Given data:

• Initial length of a spring, x0 = 0.5 m
• Final length of a spring, x = 0.2 m
• Force exerted by spring on a wooden crate, Fs = ?
• Spring constant, k = 170 N/m

Applying the formula:

• Fs = k Δx = k × (x – x0)
• Fs = 170 × (0.2 – 0.5)
• Fs = 170 × (-0.3)
• Fs = -51 N

Therefore, the force exerted by spring on a wooden crate is 51 N. (right)