A motor drives a rope drum of 820 mm diameter, which in turn drags a component of mass 650 kg, 175 m up a slope that makes an angle of 8 degrees to the horizontal. the component starts from rest and uniformly reaches a velocity of 5.3 m/s on completion of its journey. use d'alembert's principle to determine:
a) the force requires from the cable to complete the task if the coefficient of friction between the matig surfaces is 0.37
b) the input torque supplied to the drum by the motor id the mass of the drum is 105 kg and its radius of gyration is 280mm. take the frictional torque in the bearings as 6Nm
c) the work done
d) the power displaced by the motor
To solve these problems using D'Alembert's principle, we need to consider the forces acting on the component being dragged up the slope.
a) Force Required from the Cable:
First, let's calculate the force required from the cable to complete the task. We can break this force into two components: the force required to overcome gravity, and the force required to overcome friction.
The force required to overcome gravity can be calculated using the component's mass and the angle of the slope. The gravitational force can be broken down into two components: parallel to the slope (mg*sinθ) and perpendicular to the slope (mg*cosθ).
The force required to overcome friction can be calculated using the normal force (mg*cosθ) and the coefficient of friction (μ). The frictional force can be calculated as (μ * normal force).
To find the total force required, we sum the forces required to overcome gravity and friction:
Force required = Force to overcome gravity + Force to overcome friction
Force required = (mg*sinθ) + (μ * normal force)
Force required = (650 kg * 9.8 m/s^2 * sin 8°) + (0.37 * 650 kg * 9.8 m/s^2 * cos 8°)
b) Input Torque supplied to the Drum:
To calculate the input torque supplied to the drum, we need to consider the forces acting on the drum. These include the tension force in the cable, the frictional torque in the bearings, and the torque required to accelerate the drum.
The tension force in the cable can be calculated by multiplying the force required from the cable by the radius of the drum (820 mm / 2 = 410 mm).
Tension force = Force required * radius of drum
The net torque acting on the drum is the difference between the torque supplied by the tension force and the frictional torque in the bearings.
Torque supplied = Tension force * radius of drum - Frictional torque in bearings
c) Work Done:
The work done during the process is the product of the distance traveled and the force applied. In this case, the force applied is the force required from the cable (calculated in part a).
Work done = Force required * distance traveled
d) Power Displaced by the Motor:
Power is the rate at which work is done. It can be calculated by dividing the work done by the time taken.
Power = Work done / Time taken
Remember to convert the units to ensure consistency.
By using the equations described above, you can calculate the values for parts a), b), c), and d) using the given information.