Chapter 6: 8-24P (page 138)

A 52-kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. (a) What is the maximum torque she exerts? (b) How could she exert more torque?

Short Answer

Expert verified

a) The maximum torque exerted by the person is 86.63 m N.

b) She could exert more torque by pushing down the pedal with her weight and the muscles of her other leg. She could also use toe straps to pull one pedal up with the muscles of the other leg.

Step by step solution

Identification of the given data

The mass of the person is\(m = 52\;{\rm{kg}}\).

The radius of the circular motion traced by the pedal is\(r = 17{\rm{cm}}\; = 0.17\;{\rm{m}}\).

Definition of torque

Torque is defined as the measure of the force that causes an object to rotate about an axis. It is also called the rotational equivalent of linear force.

Mathematically, torque is represented as follows:

\(\begin{aligned}{c}\vec \tau = \vec r \times \vec F\\\left| \tau \right| = rF\sin \theta \end{aligned}\) … (i)

Here, Fi is the applied force, and r is the perpendicular distance between the point about which the torque and point of application of force are calculated.

(a) Determination of maximum torque

The maximum torque will be exerted by the force of her weight. It means that the direction of her weight will be tangential (making a right angle) to the axis of rotation.

\( \Rightarrow \theta = {90^{\rm{o}}}\)

From equation (1),

\(\begin{aligned}{c}\left| \tau \right| &= rF\sin \theta \\ &= r\left( {mg} \right)\sin \theta \\ &= \left( {0.17\;{\rm{m}}} \right)\left( {52\;{\rm{kg}}} \right)\left( {9.8\;{\rm{m/}}{{\rm{s}}^2}} \right)\sin {90^{\rm{o}}}\\ &= 86.63\;{\rm{m}}\;{\rm{N}}\end{aligned}\)

Thus, the maximum torque exerted by the person is 86.63 m N.