# Fig.19 shows a crane lifting a car of mass 850 kg at constant velocity through a height of 12m in a time of 40s. The crane has a working efficiency of 60%.

i) Calculate the tension in the lifting cable (in N)
ii) Calculate the total input power required by the crane to lift the car (in W)
iii) Suggest and explain many ways the crane can be modified to improve its efficiency

## I will give you the meaning of the abbreviations, but you will have to write them into the equations.

F = Force in newtons.

2. M = Mass in kilograms.

3. g = Acceleration due to gravity in m/s^2.

4. d = Distance in meters.

5. J = Joules.

6. s = Time in seconds.

7. Pin = Power input.

8. Eff. = Efficiency.

## I. F = M*g = 850 * 9.8 = 8330 N. = Tension in the cable.

II. Po = F*d/t = 8330 * 12/40 = 2499 J./s = 2499 Watts. = Power out.

Pin = Po/Eff. = 2499/0.60 = 4165 Watts.
= Input power.

## i) To calculate the tension in the lifting cable, we can use the equation: Tension = Weight + Friction.

First, let's calculate the weight of the car:
Weight = mass * gravity
Weight = 850 kg * 9.8 m/s^2 = 8330 N

Since the car is moving at a constant velocity, the net force on it is zero, which means the force due to friction is equal to the tension in the cable.

Friction = Tension

Therefore:
Tension = Weight + Friction = 2 * Weight

Tension = 2 * 8330 N = 16660 N

ii) The total input power required by the crane can be calculated using the equation: Power = Work Done / Time

The work done is equal to the force applied multiplied by the distance it acts over:
Work Done = Force * Distance

In this case, the force is equal to the tension in the cable, and the distance is the height through which the car is lifted:
Work Done = Tension * Distance = Tension * Height

Work Done = 16660 N * 12 m = 199,920 J (Joules)

Now we can calculate the total input power:
Power = Work Done / Time
Power = 199,920 J / 40 s = 4998 W (Watts)

iii) To improve the efficiency of the crane, we can suggest the following modifications:

1. Reduce friction: By using lubricants or improving the lubrication system, the amount of friction between the moving parts of the crane can be reduced. This would decrease the force due to friction and therefore increase the efficiency.

2. Upgrade the motor: By using a more powerful motor, the crane can lift the car more efficiently, reducing the required input power.

3. Optimize the design: The crane can be modified to have a more aerodynamic shape or reduce unnecessary weight. These modifications would reduce air resistance and make the crane more efficient.

4. Improve the pulley system: By using higher-quality pulleys with lower friction, the efficiency of the crane can be improved.

5. Use lighter materials: Constructing the crane using lighter materials can reduce the weight that needs to be lifted, leading to increased efficiency.

Remember, improving the efficiency of a crane involves a combination of reducing friction, optimizing design, upgrading components, and using more efficient materials and systems.

## i) To calculate the tension in the lifting cable, we can use the equation:

Force = mass * acceleration

Since the car is being lifted at constant velocity, the acceleration is zero. Therefore, the force needed to lift the car is equal to its weight.

Weight = mass * gravity

where gravity is the acceleration due to gravity (approximately 9.8 m/s^2).

Weight = 850 kg * 9.8 m/s^2 = 8330 N

However, since the crane has a working efficiency of 60%, we need to account for the fact that only 60% of the input energy is converted into useful work. So, we divide the calculated weight by the efficiency:

Tension in lifting cable = (Weight / efficiency) = (8330 N / 0.6) = 13883.33 N (to 2 decimal places)

ii) The total input power required by the crane can be calculated using the formula:

Power = work done / time

The work done by the crane is equal to the force applied (tension in the lifting cable) multiplied by the distance covered (height).

Work done = force * distance

Work done = (Tension in lifting cable) * (distance)

Work done = 13883.33 N * 12 m = 166599.96 J (to 2 decimal places)

Now, we can calculate the total input power using the formula:

Power = work done / time

Power = 166599.96 J / 40 s = 4164.999 W (to 3 decimal places)

iii) To improve the efficiency of the crane, several modifications can be made:

1. Reduce friction: Minimizing friction in the crane's moving parts by proper lubrication and maintenance can reduce energy losses.

2. Increase mechanical advantage: Increasing the mechanical advantage of the crane's pulley system or using a more efficient lifting mechanism can reduce the force required, leading to lower energy consumption.

3. Streamline design: Reducing air resistance by designing the crane with aerodynamics in mind can help improve efficiency.

4. Improve motor efficiency: Upgrading to a more efficient motor can reduce energy losses and improve the overall efficiency of the crane.

5. Optimize cable length: Adjusting the length of the lifting cable to minimize unnecessary weight and friction can improve the crane's efficiency.

6. Use lightweight materials: Constructing the crane using lightweight, strong materials can reduce its overall weight and the force required to lift loads.

By implementing these modifications, the crane can become more energy-efficient and require less power to perform the same tasks.