A spring lies on a horizontal table, and the left end of the spring is attached to a wall. The other end is connected to a box. The box is pulled to the right, stretching the spring. Static friction exists between the box and the table, so when the spring is stretched only by a small amount and the box is released, the box does not move. The mass of the box is 0.75 kg, and the spring has a spring constant of 60 N/m. The coefficient of static friction between the box and the table on which it rests is μs = 0.58. How far can the spring be stretched from its unstrained position without the box moving when it is released?

Question

A spring lies on a horizontal table, and the left end of the spring is attached to a wall. The other end is connected to a box. The box is pulled to the right, stretching the spring. Static friction exists between the box and the table, so when the spring is stretched only by a small amount and the box is released, the box does not move. The mass of the box is 0.75 kg, and the spring has a spring constant of 60 N/m. The coefficient of static friction between the box and the table on which it rests is μs = 0.58. How far can the spring be stretched from its unstrained position without the box moving when it is released?

Answer 1

To determine how far the spring can be stretched without the box moving, we need to find the maximum force of static friction acting on the box. Once the force exerted by the spring exceeds this maximum static friction, the box will start to move.

First, let's calculate the maximum static friction force using the coefficient of static friction and the weight of the box.

1. Calculate the weight of the box:
The weight of the box is given by the formula:
Weight = mass * acceleration due to gravity

Weight = 0.75 kg * 9.8 m/s^2 (acceleration due to gravity)
Weight = 7.35 N

2. Calculate the maximum static friction force:
Maximum static friction force = coefficient of static friction * Normal force

Since the box is on a horizontal table and is not moving vertically, the normal force is equal to the weight of the box.

Maximum static friction force = 0.58 * 7.35 N
Maximum static friction force = 4.258 N (approximately)

Now, let's determine the maximum displacement of the spring without the box moving. At this displacement, the force exerted by the spring will equal the maximum static friction force.

3. Calculate the maximum displacement of the spring:
According to Hooke's Law, the force exerted by a spring is given by the formula:
Force (F) = spring constant (k) * displacement (x)

At the maximum displacement x, the force exerted by the spring should be equal to the maximum static friction force:

k * x = maximum static friction force

Rearranging the equation, we can solve for x:

x = maximum static friction force / spring constant

x = 4.258 N / 60 N/m (spring constant)
x = 0.07097 m (approximately)

Therefore, the spring can be stretched a maximum of approximately 0.07097 meters without the box moving when it is released.