Max Breaking Force on Car & Friction of Mop: Coefficient, Mass, Angle

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The coefficient of static friction for car tires on a dry road is 0.62, allowing for a maximum braking force calculation based on the car's mass of 1600 kg. On a level road, the maximum braking force can be determined using the formula F = μ * m * g, resulting in a force of approximately 9,792 N. On an 8.6° downgrade, the braking force is reduced due to the gravitational component acting down the slope, necessitating adjustments in calculations. For the mop scenario, the angle θ of the handle affects the force required to slide the mop, with a critical angle θ0 determining the threshold for motion. If θ is less than θ0, the mop will not slide regardless of the applied force.
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The coefficient of static friction between the tires of a car and dry road is 0.62. The mass of the car is 1600 kg. What maximum breaking force is obtainable (a) on a level road and (b) 8.6° downgrade?

The handle of a floor mop of mass m makes an angle θ with the vertical direction. Let μk be the coefficient of kinetic friction between mop and floor and μs the coefficient of static friction between mop and floor. Neglect the mass of the handle. (a) Find the magnitude of the force F directed along the handle required to slide the mop with uniform velocity across the floor. (b) Show that θ is smaller than a certain θ0 of the mop cannot be made to slide across the floor no matter how great a force is directed along the handle. What is angle θ0?
 
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