a 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 the step, we need to consider the gravitational force acting on the roller and the force required to overcome the vertical component of the step.

Let's break down the problem step-by-step:

Step 1: Calculate the weight of the roller
The weight of an object is given by the formula: Weight = mass × acceleration due to gravity (g).

Given:
Mass of the roller (m) = 450 kg
Acceleration due to gravity (g) = 9.8 m/s^2

Weight of the roller (W) = m × g
W = 450 kg × 9.8 m/s^2

Step 2: Calculate the vertical component of the step
The vertical component of the step is given by: Vertical component = step height × sin(angle)

Given:
Step height = 250 mm = 0.25 m
Angle (θ) = 30°

Vertical component (V) = 0.25 m × sin(30°)

Step 3: Calculate the minimum force required to pull up the step
The minimum force required can be found using the equation: Force = Weight + Vertical component

Force = W + V

Substitute the values from Step 1 and Step 2 into the equation:

Force = (450 kg × 9.8 m/s^2) + (0.25 m × sin(30°))

Now you can calculate the minimum force required to pull up the step.

To find the minimum force required to pull the roller up the step, we need to consider the weight of the roller and the angle at which it is being pulled.

Let's break down the problem step by step:

Step 1: Calculate the weight of the roller.
The weight of an object is given by the formula W = m * g, where W is the weight, m is the mass, and g is the acceleration due to gravity (approximately 9.8 m/s²).

Given that the mass of the roller is 450 kg, the weight can be calculated as follows:
W = 450 kg * 9.8 m/s²
W = 4410 N

Step 2: Determine the vertical component of the force.
The force required to lift the roller up the step consists of two components: the vertical component and the horizontal component.

Since the step is vertical, the vertical component of the force is equal to the weight of the roller, which we calculated in Step 1:
Vertical component = W = 4410 N

Step 3: Determine the horizontal component of the force.
To calculate the horizontal component, we need to use trigonometry. We know that the handle of the roller makes an angle of 30 degrees with the horizontal.

The horizontal component can be determined using the formula:
Horizontal component = Force * cos(angle)

The angle is given as 30 degrees, so we can substitute the values and calculate the horizontal component:
Horizontal component = Force * cos(30) [Note: Angle should be in radians]
Horizontal component = Force * cos(30°) [Use the conversion: cos(π/6) = √3/2]

Step 4: Use the vertical and horizontal components to find the minimum force.
To pull the roller up the step, we need a force that has both a vertical and a horizontal component. The minimum force required can be calculated using the Pythagorean theorem:

Force = √(vertical component² + horizontal component²)

Substituting the values we found in Steps 2 and 3:
Force = √(4410² + (Force * cos(30))^2)

Now, we can solve this equation to find the minimum force required to pull the roller up the step.

However, it seems that there is a circular dependency on the "Force" variable in the equation. It cannot be solved algebraically. We may need to use numerical methods such as trial and error or iteration to find the solution.