Question

A man weighing 180 lbf is pushing a cart that weighs 100 lbf with its contents up a ramp that is inclined at an angle of \(10^{\circ}\) from the horizontal. Determine the work needed to move along this ramp a distance of \(100 \mathrm{ft}\) considering \((a)\) the \(\operatorname{man}\) and \((b)\) the cart and its contents as the system. Express your answers in both lbf.ft and Btu.

Short answer

Question: Calculate the work required to push a man weighing 180 lbf and a cart weighing 100 lbf up a 100 ft ramp inclined at 10 degrees. Answer: The work done to move the system along the ramp is approximately 4854.7 lbf.ft or 6.24 Btu.

Step-by-step solution

Find the total weight of the system

We have the weight of the man as 180 lbf and the weight of the cart with its content as 100 lbf. Let's find the total weight of the system. Total weight (W_total) = Weight of man (W_man) + Weight of cart with contents (W_cart) W_total = 180 lbf + 100 lbf = 280 lbf

Calculate the force component parallel to the ramp

We need to find the force acting parallel to the ramp's surface to calculate the work. To do this, we consider the angle of the ramp, which is given as \(10^{\circ}\). Force parallel to the ramp (F_parallel) = Total weight * sin(Inclination angle) Inclination angle = \(10^{\circ}\). To convert it into radians, we multiply by \(\frac{\pi}{180}\). Inclination angle (theta) = \(10 \times \frac {\pi} {180} = \frac{\pi}{18}\) radians F_parallel = 280 lbf * sin(\(\frac{\pi}{18}\))

Calculate the work done to move the system along the ramp

Now that we have the force acting parallel to the ramp, we can calculate the work done to move the system along the ramp. Work done (W) = Force parallel to the ramp (F_parallel) * Distance along the ramp (d) Distance along the ramp (d) = 100 ft W = F_parallel * 100 ft

Convert work done from lbf.ft to Btu

To convert the work done from lbf.ft to Btu, we use the conversion factor 1 Btu = 778.169 lbf.ft. Work done in Btu (W_Btu) = Work done in lbf.ft (W) / 778.169 Substitute the values obtained in previous steps to get our final answer. Step 1: W_total = 280 lbf Step 2: F_parallel = 280 lbf * sin(\(\frac{\pi}{18}\)) Step 3: W = F_parallel * 100 ft Step 4: W_Btu = W / 778.169 Calculate these values and express the final answer for the work done in lbf.ft and Btu.