A roller of mass 300kg and radius 50 cm lying on horizontal floor is resting against step of height 20 cm. the minimum horizontal force to be applied on the roller on the step is
Ans:3920N
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To find the minimum horizontal force required to move the roller over the step, we need to consider the forces acting on the roller. In this case, we have:
1. Weight force (mg): The weight of the roller exerts a downward force due to gravity. The magnitude of this force is given by mg, where m is the mass of the roller and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Normal force (N): When the roller is resting on the horizontal floor, the surface exerts an equal and opposite force called the normal force. This force acts perpendicular to the surface and prevents the roller from sinking into the floor.
3. Friction force (f): As the roller rests against the step, there is a friction force acting between the roller and the floor. This force opposes the motion of the roller and needs to be overcome to lift the roller onto the step.
To find the minimum horizontal force, we need to calculate the friction force. The friction force can be found using the equation:
f = μN
where μ is the coefficient of friction between the roller and the floor, and N is the normal force.
The normal force can be found using the weight force:
N = mg
Now, we need to find the coefficient of friction. In this case, the friction force is needed to lift the roller onto the step. This can be considered as static friction, which is the maximum friction force that can be exerted before the object starts to move.
The equation for static friction is:
fs ≤ μsN
where μs is the coefficient of static friction. We can rearrange this equation to find the coefficient of static friction:
μs ≥ fs/N
Given that the step height is 20 cm (0.2 m) and the radius of the roller is 50 cm (0.5 m), the angle between the floor and the step can be calculated as:
θ = atan(step height / roller radius)
θ = atan(0.2 / 0.5)
θ ≈ 0.377
Now, the coefficient of static friction can be found using this angle:
μs = tan(θ)
μs ≈ 0.3907
Finally, we can substitute the values into the equation to find the friction force:
f = μN
f = μsN
f = μs * mg
f = 0.3907 * (300 kg * 9.8 m/s^2)
f ≈ 1146.66 N
Therefore, the minimum horizontal force required to move the roller onto the step is approximately 1146.66 N.