# Spring force

Spring force is the force exerted by a compressed or stretched spring upon any object that is attached to it. This force acts to return the spring to its natural length, pulling objects back when stretched and pushing them back when compressed. For example, when you stretch a spring by pulling its ends apart, you feel the spring pulling back towards its original position. Conversely, when you compress the spring by pushing its ends together, it pushes back against the compression. This force is at play in various everyday objects, such as the suspension systems in vehicles, which absorb shocks from the road, and in mechanical pens, where the spring enables the pen to retract and extend.

The equation for spring force, described by Hooke’s law, is given by:

$$F = -kx$$

where F is the spring force, k is the spring constant (a measure of the stiffness of the spring), and x is the displacement of the spring from its equilibrium position. The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement, acting to restore the spring to its natural length.

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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)