Question

How long will it take a 2750-W motor to lift a 385-kg piano to a sixth-story window 16.0 m above?

Short answer

The time taken to lift the piano to a sixth story window is 22 s.

Step-by-step solution

Definition of power

Power P is defined as the rate at which work is done and is obtained by dividing the time taken to perform that work.

Power can, mathematically, be expressed as:

\(P = \frac{{{\rm{Work}}}}{{{\rm{Time}}}} = \frac{W}{t}\)

Identification of the given data

Theaverage output power of the motor is \(P = 2750\;{\rm{W}}\).

The mass of the piano is \(m = 385\;{\rm{kg}}\).

The height from the ground is \(h = 16\;{\rm{m}}\).

Determination of the work done

When an object is lifted against gravity, it requires gravitational potential energy. Here, the work done will be equal to the potential energy.

\(\begin{aligned}W &= mgh\\ &= \left( {385\;{\rm{kg}}} \right)\left( {9.8\;{\rm{m/}}{{\rm{s}}^2}} \right)\left( {16\;{\rm{m}}} \right)\\ &= 60368\;{\rm{J}}\end{aligned}\)

Determination of the time taken

From the expression of power, the time taken is determined as:

\(\begin{aligned}t &= \frac{W}{P}\\ &= \frac{{60368\;{\rm{J}}}}{{2750\;{\rm{W}}}}\\ &= 21.95\;{\rm{s}}\\ &\approx 22\;{\rm{s}}\end{aligned}\)

Thus, the time taken to lift the piano is 22 s.