A cricket pitch roller of diameter 1.5 m has a mass of 450 kg; it is pulled by a handle attached to its centre. Find the minimum force to pull up a step which is 250 mm high if the roller handle makes an angle of 30o with the horizontal.
To find the minimum force required to pull up a step with the cricket pitch roller, we need to consider the forces acting on the roller.
Let's break down the problem and analyze it step by step:
Given:
- Diameter of the roller (d) = 1.5 m
- Mass of the roller (m) = 450 kg
- Height of the step (h) = 250 mm = 0.25 m
- Angle of the roller handle with the horizontal (θ) = 30°
First, let's calculate the weight of the roller:
Weight (W) = mass (m) * acceleration due to gravity (g)
W = m * g
where g ≈ 9.8 m/s^2 (acceleration due to gravity)
W = 450 kg * 9.8 m/s^2
W = 4410 N
Next, let's determine the vertical component of the weight that needs to be overcome to lift the roller onto the step. This component can be found using trigonometry:
Vertical component of weight (Wv) = weight (W) * sin(θ)
Wv = W * sin(θ)
Wv = 4410 N * sin(30°)
Now, let's calculate the force required to overcome the vertical component of the weight:
Force required (F) = Wv + additional force due to step (Fstep)
Since we want to find the minimum force required, we assume that the roller is lifted with just enough force to overcome the vertical component of the weight:
F = Wv
Substituting the value of Wv:
F = 4410 N * sin(30°)
F = 4410 N * 0.5
F = 2205 N
Therefore, the minimum force required to pull up the cricket pitch roller over a step that is 250 mm high, with the roller handle at an angle of 30° with the horizontal, is 2205 Newtons (N).