Chapter 6, Problem 45P

Part (a)

To determine

The amount of work done against gravity.

Answer to Problem 45P

Solution: $\text{9}{\text{.19×10}}^{\text{4}}\text{J}$

Explanation of Solution

Given:

Inclination of hill = $\text{7}\text{.50}°$

Height of hill $=125\text{ cm}$

Diameter of pedals $=36.0\text{ cm}$

Mass of bicycle and person $=75.0\text{ kg}$

Formula Used:

Work done against gravity = Change in Potential energy

  $\Delta PE=mg{h}_2-mg{h}_1$

Calculation:

Initial potential energy

  $\begin{array}{l}=mg{h}_1\\ \text{=75}\text{.0}\text{kg×9}\text{.80}{\text{m/s}}^{\text{2}}\text{×0}\text{m}\\ \text{=0}\text{J}\end{array}$

Final potential energy

  $\begin{array}{l}=mg{h}_2\\ \text{=75}\text{.0}\text{kg×9}\text{.80}{\text{m/s}}^{\text{2}}\text{×125}\text{m}\\ \text{=91875}\text{J}\end{array}$

  $\begin{array}{c}\Delta P.E.=P{\text{.E}}_{\text{2}}\text{-P}{\text{.E}}_{\text{1}}\\ =\text{(91875-0)}\text{J}\\ =\text{9}{\text{.19×10}}^{\text{4}}\text{J}\end{array}$

Conclusion: Hence, the work done against gravity= $\text{9}{\text{.19×10}}^{\text{4}}\text{J}$

Part (b)

To determine

Average force that must be exerted on the pedals tangent to their circular path.

Answer to Problem 45P

Solution: 433 N

Explanation of Solution

Given:

Inclination of hill = $\text{7}\text{.50}°$

Height of hill $=125\text{ cm}$

Diameter of pedals $=36.0\text{ cm}$

Mass of bicycle and person $=75.0\text{ kg}$

Distance moved in one complete revolution = 5.10 m

Formula Used:

Work done by cyclist in one revolution = tangential force on pedals $\times$ circumference of pedals circle

Calculation:

Work done against gravity in one revolution = Change in potential energy in one revolution

  $\frac{\text{Total change in PE}}{\text{Total height change}}\text{×height change in one revolution}$

Change in height in one revolution:

  $\begin{array}{l}\text{=}\left(\text{5}\text{.10 m}\right)\text{×}\left(\text{sin 7}{\text{.50}}^{\text{0}}\right)\\ \text{=0}\text{.666 m}\end{array}$

Work done against gravity in one revolution

  $\begin{array}{l}=\frac{\text{91875J}}{\text{125}\text{m}}\text{×0}\text{.666}\text{m}\\ \text{=489}\text{.51}\text{J}\end{array}$

Circumference of pedals = $\text{π × diameter of pedals}$

  $\text{= π×0}\text{.36m}$

= $\text{1}\text{.13m}$

Work done by cyclist in one revolution = tangential force on pedals $\times$ circumference of pedals circle

  $\text{489}\text{.51}\text{J=tangential force on pedals ×1}\text{.13}\text{m}$

Tangential on pedals

  $\begin{array}{l}\text{=}\frac{\text{489}\text{.51}\text{J}}{\text{1}\text{.13}\text{m}}\\ \text{=}\text{433 N}\end{array}$

Conclusion: Hence, tangential force acting on the pedals = $\text{433N}$ .