Q80P - University Physics with Modern Physics

Question

A physics professor is pushed up a ramp inclined upward at 30.0° above the horizontal as she sits in her desk chair, which slides on frictionless rollers. The combined mass of the professor and chair is 85.0 kg. She is pushed 2.50 m along the incline by a group of students who together exert a constant horizontal force of 600 N. The professor’s speed at the bottom of the ramp is 2.00 m/s. Use the work–energy theorem to find her speed at the top of the ramp.

Step-by-Step Solution

Answer

The speed at the top f the ramp is 3.17 m/s .

Step 1 :Given Data

The combined mass = mass = 85.0 N

Horizontal force =  600 N

Speed = 2.00 m / s 

Step 2: Introduction

The work-energy theorem states that the net work done by the forces on an object equals the change in its kinetic energy.

 

Work energy theorem is given by the expression as follows:

 

$W_{tot}=K_2-K_1$ 

 

Here, $W_{tot}$ is the total work done, $K_1$,$K_2$ are the initial and final kinetic energies.

Step 3: Find the speed at the top of the ramp

Calculate the speed as follows:

Substitute $\left(\mathrm{F} \mathrm{cos\theta }-\mathrm{Mg} \mathrm{sin\theta }\right)\mathrm{d}=\frac{1}{2}{\mathrm{mv}}_{\mathrm{f}}^2-\frac{1}{2}{\mathrm{mv}}^2$f     or  ${\mathrm{K}}_1$ in the equation.

$\left(F \cos \theta -Mg\sin \theta \right)d=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}^2$ 

 

Substitute 600 N for $\mathrm{F},30° \mathrm{for} \mathrm{\theta }, 85\mathrm{kgfor} \mathrm{m},9.8\mathrm{m}/{\mathrm{s}}^2 \mathrm{for},2.5 \mathrm{m} \mathrm{for} \mathrm{d} \mathrm{and} 2.0 \mathrm{m}/\mathrm{s} \mathrm{for} \mathrm{v}$.

$$\begin{gathered} \left(\left(\left(600\mathrm{N}\right)\cos 30°\right)-\left(\left(85\mathrm{kg}\right)\left(9.8 \mathrm{m}/{\mathrm{s}}^2\right)\sin 30°\right)\right)\left(2.5\mathrm{m}\right)=\frac{1}{2}\left(85 \mathrm{kg}\right)\left({\mathrm{v}}_{\mathrm{f}}^2-\left(2.0 \mathrm{m}/{\mathrm{s}}^2\right)\right) \\ {\mathrm{v}}_{\mathrm{f}}=3.17\mathrm{m}/\mathrm{s} \end{gathered}$$ 

Hence, the speed atthe top f the ramp is  3.17 m/s.