A 3.30-kN piano is lifted by three workers at a constant speed to an apartment 28.8 m above the street using a pulley system fastened to the roof of the building. Each worker is able to deliver 197 W of power, and the pulley system is 75.0% efficient (so that 25.0% of the mechanical energy is lost due to friction in the pulley). Neglecting the mass of the pulley, find the time required to lift the piano from the street to the apartment.
mg=3300 N, h=28.8 m, Pₒ=197 W
PE =mgh
E= mgh/0.75
P=E/t = mgh/0.75•t = 3•Pₒ
t= mgh/0.75•3• Pₒ =
=3300•28.8/0.75•3•197 =
= 214 s =3.57 min
To find the time required to lift the piano, we can use the concept of power.
First, let's find the total power available from the three workers. Since each worker can deliver 197 W of power, the total power available is 3 * 197 W = 591 W.
We know that power is defined as work done divided by time, and work done is equal to the force exerted times the distance traveled.
Given that the distance traveled is 28.8 m and the force exerted is 3.30 kN (or 3.30 * 1000 N), we can calculate the work done as follows:
Work done = force * distance = (3.30 * 1000 N) * 28.8 m = 95040 N·m
Since power is also equal to work done divided by time, we can rearrange the equation to find the time:
Time = Work done / Power
However, we need to account for the efficiency of the pulley system. Since 25.0% of the mechanical energy is lost due to friction in the pulley, the work done by the workers has to be more than the actual work done on the piano.
To find the actual work done on the piano, we can divide the work done by the efficiency of the pulley system:
Actual work done = Work done / Efficiency = 95040 N·m / 0.75 = 126720 N·m
Finally, we can substitute the actual work done and the total power available to find the time:
Time = Actual work done / Power = 126720 N·m / 591 W ≈ 214.47 seconds
Therefore, the time required to lift the piano from the street to the apartment is approximately 214.47 seconds.
To find the time required to lift the piano from the street to the apartment, we can use the concept of work and power.
First, let's calculate the work done to lift the piano. The work done (W) is given by the product of force (F) and distance (d) traveled. In this case, the force is the weight of the piano (3.30 kN) and the distance is the height of the apartment (28.8 m).
W = F * d
But the given force (3.30 kN) is already in kilonewtons, so we don't need to convert it further. Therefore, the work done is:
W = 3.30 kN * 28.8 m
W = 95.04 kJ (kilojoules)
Now, let's calculate the actual work done by the workers, accounting for the efficiency of the pulley system. The actual work done (Wa) by the workers is given by:
Wa = W / Efficiency
Given that the efficiency of the pulley system is 75.0% (or 0.75), we have:
Wa = 95.04 kJ / 0.75
Wa = 126.72 kJ (kilojoules)
Since power (P) is defined as the rate at which work is done over time, we can calculate the time (T) required to do the work by rearranging the equation:
P = W / T
T = W / P
But we need to consider the power delivered by each worker, so the total power (P) is the sum of the power delivered by each worker. Given that each worker can deliver 197 W of power, and there are three workers, we have:
P = 3 * 197 W
P = 591 W (watts)
Now, we can calculate the time required:
T = 126.72 kJ / 591 W
T = 0.2145 hours (or 12.87 minutes)
Therefore, it would take approximately 12.87 minutes to lift the piano from the street to the apartment.