A pulley system has an efficiency of 78.0 per-

cent.
The acceleration of gravity is 9.81 m/s2 .
How much of the rope must be pulled in if
a force of 469 N is needed to lift a 52 kg desk
2.78 m?

To solve this problem, we can use the concept of work and mechanical advantage.

1. First, let's find the work done in lifting the desk. The work done is equal to the force applied multiplied by the distance over which the force is applied. Therefore, the work done (W) can be calculated as:

W = force × distance
W = 469 N × 2.78 m
W = 1304.02 N·m

2. The mechanical advantage (MA) of a pulley system is the ratio of the output force to the input force. The efficiency (ε) of the pulley system is given as 78.0%. Efficiency is defined as the ratio of the actual output work to the input work. Therefore, the efficiency can be calculated as:

ε = (output work / input work) × 100
78.0 = (output work / input work) × 100
output work / input work = 78.0 / 100
output work / input work = 0.78

As the input work is equal to the work done in lifting the desk (1304.02 N·m), the output work can be calculated as:

output work = 0.78 × 1304.02 N·m
output work ≈ 1016.63 N·m

3. The mechanical advantage can also be calculated as the ratio of the distance the rope is pulled to the distance the desk is lifted. Let's assume the distance the rope is pulled is d:

MA = distance pulled / distance lifted
MA = d / 2.78

4. Substitute the values of the output work (1016.63 N·m) and mechanical advantage (MA) into the work formula:

output work = input force × output distance
1016.63 N·m = 469 N × (d / 2.78)

5. Solve for the distance the rope must be pulled (d):

d = (1016.63 N·m × 2.78) / 469 N
d ≈ 6.04 m

Therefore, approximately 6.04 m of the rope must be pulled to lift the 52 kg desk 2.78 m using a force of 469 N in a pulley system with an efficiency of 78.0%.

To solve this problem, we can use the concept of work and mechanical advantage. First, let's solve for the work done to lift the desk.

The work done (W) is given by the equation W = force * distance. In this case, the force is 469 N and the distance is 2.78 m. Substituting these values into the equation, we get:

W = 469 N * 2.78 m
W = 1303.82 N·m

Next, let's calculate the mechanical advantage (MA) of the pulley system. The mechanical advantage represents the ratio of the output force to the input force in a machine system. In this case, the input force is the force needed to lift the desk, which is 469 N. The output force is the force that the person pulling the rope needs to apply. The mechanical advantage (MA) can be calculated using the formula:

MA = Output force / Input force

Since the efficiency of the system is given as 78.0 percent, we need to convert this to a decimal by dividing it by 100.

Efficiency = 78.0 / 100 = 0.78

The output force can be calculated by multiplying the input force by the efficiency:

Output force = Input force * Efficiency
Output force = 469 N * 0.78
Output force = 365.82 N

Now, let's calculate the mechanical advantage:

MA = Output force / Input force
MA = 365.82 N / 469 N
MA = 0.78

The mechanical advantage represents the ratio of the length of rope pulled to the distance the desk is lifted. Therefore, we can now set up the equation to solve for the length of rope pulled:

MA = Length of rope pulled / Distance lifted

Rearranging the equation, we get:

Length of rope pulled = MA * Distance lifted
Length of rope pulled = 0.78 * 2.78 m
Length of rope pulled = 2.1674 m

So, approximately 2.1674 meters of the rope must be pulled in order to lift the 52 kg desk a distance of 2.78 meters.

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