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Mechanics: Static Friction, acceleration and circular motion

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1.1 Acceleration
a. The intent of the problem is to determine the maximum acceleration of the train in which a box lying on its floor would remain stationary. Assume the coefficient of static friction between the block and the train's floor to be 0.15. See Fig.1

1.2 Static Friction
a. See Fig. 2 in the attached document where mass of 5 kg rests on a horizontal plane. The incline of the plane is gradually increased until at 15 degrees w.r.t. the horizontal plane, the mass begins to slide. What is the coefficient of static friction between the mass and the surface of the plane?

1.3 Circular Motion
a. Imagine a cyclist riding at a speed of 18km/h on a leveled road and takes a sharp circular turn of radius 3 meters without compromising the speed. Would cyclist be at a risk of slipping while taking this turn, if the coefficient of static friction between the tires and the road is 0.1 (See Fig. 3)?

b. A circular racetrack of radius 200 m is banked at an angle of 15 degrees. At what speed should the car be driven in order to avoid the wear and tear on the tires and maximum permissible speed to avoid slipping. Assume the coefficient of static friction to be 0.2. See Fig. 4

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Solution 1.1 a

The box in the given problem is stationary because of the frictional force acting on it despite the acceleration of the train. As we keep on increasing the acceleration of the
train, the frictional force acting on the box will reach a limiting value, after which the box starts sliding.

In order to determine the maximum accelaration of the train at which the box remains stationary, we'll have to calculate the frictional force which is dependent upon
the reaction force from the floor of the train.
Mathematically speaking,
fs ≤ µsN = µs mg
Also, as per Newton's second law:
the force acting on the object is the product of it's mass and accelaration. Mathematically putting it for the frictional force above, we get
fs= ma

where,
µs is the coefficient of static friction
a is the accelaration
m is the pass
g is the constant (9.81 m/s2)

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