A car is negotiating a curve of radius 100 m at a speed of 15 m/s. What is the minimum coefficient of friction required to prevent the car from skidding?
Practice Questions
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Q1
A car is negotiating a curve of radius 100 m at a speed of 15 m/s. What is the minimum coefficient of friction required to prevent the car from skidding?
0.15
0.25
0.30
0.35
Frictional force = m * a_c; μmg = mv²/r; μ = v²/(rg) = (15 m/s)² / (100 m * 9.8 m/s²) ≈ 0.23.
Questions & Step-by-step Solutions
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Q
Q: A car is negotiating a curve of radius 100 m at a speed of 15 m/s. What is the minimum coefficient of friction required to prevent the car from skidding?
Solution: Frictional force = m * a_c; μmg = mv²/r; μ = v²/(rg) = (15 m/s)² / (100 m * 9.8 m/s²) ≈ 0.23.
Steps: 8
Step 1: Identify the given values. The radius of the curve (r) is 100 meters, the speed of the car (v) is 15 meters per second, and the acceleration due to gravity (g) is approximately 9.8 meters per second squared.
Step 2: Understand that the car needs a frictional force to keep it from skidding while going around the curve. This frictional force can be expressed as the product of the coefficient of friction (μ), the mass of the car (m), and the acceleration due to gravity (g).
Step 3: The centripetal acceleration (a_c) required to keep the car moving in a circle is given by the formula a_c = v² / r. Plug in the values: a_c = (15 m/s)² / (100 m).
Step 4: Calculate the centripetal acceleration: a_c = 225 m²/s² / 100 m = 2.25 m/s².
Step 5: Set up the equation for the frictional force: μmg = ma_c. Since mass (m) appears on both sides, we can simplify it to μg = a_c.
Step 6: Rearrange the equation to solve for the coefficient of friction (μ): μ = a_c / g.
Step 7: Substitute the values into the equation: μ = 2.25 m/s² / 9.8 m/s².
Step 8: Calculate the coefficient of friction: μ ≈ 0.229, which we can round to approximately 0.23.
