A cyclist is negotiating a circular turn of radius 30 m at a speed of 15 m/s. Wh
Practice Questions
Q1
A cyclist is negotiating a circular turn of radius 30 m at a speed of 15 m/s. What is the minimum coefficient of friction required to prevent slipping?
0.25
0.5
0.75
1
Questions & Step-by-Step Solutions
A cyclist is negotiating a circular turn of radius 30 m at a speed of 15 m/s. What is the minimum coefficient of friction required to prevent slipping?
Step 1: Identify the given values. The radius of the turn (r) is 30 meters, the speed of the cyclist (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 when a cyclist turns, they need a centripetal force to keep moving in a circle. This force comes from friction between the tires and the ground.
Step 3: The formula for centripetal acceleration (a_c) is a_c = v² / r. Calculate a_c using the given values: a_c = (15 m/s)² / (30 m).
Step 4: Calculate a_c: (15 m/s)² = 225 m²/s², and then divide by 30 m: a_c = 225 / 30 = 7.5 m/s².
Step 5: The frictional force (F_friction) needed to provide this centripetal acceleration is given by F_friction = m * a_c, where m is the mass of the cyclist and bike.
Step 6: The frictional force can also be expressed as F_friction = μ * m * g, where μ is the coefficient of friction and g is the acceleration due to gravity.
Step 7: Set the two expressions for frictional force equal to each other: μ * m * g = m * a_c. The mass (m) cancels out from both sides.
Step 8: Rearrange the equation to solve for μ: μ = a_c / g.
Step 9: Substitute the values for a_c and g: μ = 7.5 m/s² / 9.8 m/s².
Step 10: Calculate μ: μ ≈ 0.7653. This means the minimum coefficient of friction required to prevent slipping is approximately 0.765.
Centripetal Force – The force required to keep an object moving in a circular path, directed towards the center of the circle.
Friction – The force that opposes the relative motion of two surfaces in contact, which is necessary to prevent slipping in this scenario.
Coefficient of Friction – A dimensionless scalar value that represents the frictional force between two bodies in contact.
