You need to move a 129-kg sofa to a different location in the room. It takes a force of 90 N to start it moving. What is the coefficient of static friction between the sofa and the carpet?
To find the coefficient of static friction between the sofa and the carpet, we can use the formula for static friction:
\(f_s = \mu_s \cdot N\)
Where:
\(f_s\) is the force of static friction
\(\mu_s\) is the coefficient of static friction
\(N\) is the normal force
In this problem, the normal force is equal to the weight of the sofa, which can be calculated using the formula:
\(N = m \cdot g\)
Where:
\(m\) is the mass of the sofa
\(g\) is the acceleration due to gravity, which is approximately 9.8 m/s².
Given that the mass of the sofa is 129 kg and the force needed to start it moving is 90 N, we can calculate the coefficient of static friction.
Step 1: Calculate the weight of the sofa
\(N = m \cdot g = 129 \, \text{kg} \cdot 9.8 \, \text{m/s²} = 1264.2 \, \text{N}\)
Step 2: Calculate the coefficient of static friction
\(f_s = \mu_s \cdot N\)
\(90 \, \text{N} = \mu_s \cdot 1264.2 \, \text{N}\)
To isolate \(\mu_s\):
\(\mu_s = \frac{90 \, \text{N}}{1264.2 \, \text{N}}\)
Step 3: Calculate the coefficient of static friction
\(\mu_s = 0.071 \, \text{to} \, 0.072\)
Therefore, the coefficient of static friction between the sofa and the carpet is approximately 0.071 to 0.072.
To find the coefficient of static friction between the sofa and the carpet, we can use the equation:
\[ f_{\text{{static}}} = \mu_s \times m \times g \]
where:
\( f_{\text{{static}}} \) is the force of static friction,
\( \mu_s \) is the coefficient of static friction,
\( m \) is the mass of the sofa, and
\( g \) is the acceleration due to gravity (approximately 9.8 m/s²).
In this case, we know the mass of the sofa is 129 kg and the force required to start it moving is 90 N.
Since the force required to start the sofa moving is the maximum force of static friction, it means \( f_{\text{{static}}} = 90 \) N.
Let's substitute all the values we know into the equation:
\[ 90 = \mu_s \times 129 \times 9.8 \]
To solve for \( \mu_s \), we divide both sides of the equation by \( 129 \times 9.8 \):
\[ \mu_s = \frac{{90}}{{129 \times 9.8}} \]
Calculating this, we get:
\[ \mu_s \approx 0.069 \]
Therefore, the coefficient of static friction between the sofa and the carpet is approximately 0.069.