Q: What is the ratio of the biceps force to the combined weight of the arm and the book in the scenario above?

$$\text{Ratio} = \frac{F_{biceps}}{(m_{book} + m_{arm})g} = \frac{456}{63.7} \approx 7.2$$

Q: A cyclist applies a force to the pedal, generating a counter-clockwise torque of $$140\ \mathrm{Nm}$$ when the crank is at $$90^\circ to $$the vertical. What is the torque generated when the crank is at $$75^\circ to $$the vertical?

$$\tau = 140 \sin(75^\circ) = 135.3\ \mathrm{Nm}$$

Q: For the same cyclist, what is the torque generated when the crank is at $$180^\circ to $$the vertical?

$$\tau = 140 \sin(180^\circ) = 0\ \mathrm{Nm}$$

Question 1

A student holds a 4.0 kg physics book in her hand as shown in the diagram, with her forearm horizontal and the biceps muscle pulling vertically upward at a distance of 4 cm from the elbow. The center of mass of the forearm (mass 2.5 kg) is 16 cm from the elbow, and the book is 38 cm from the elbow. What is the force the biceps muscle must exert to keep the arm in equilibrium? (Assume $$g = 9.8\ \mathrm{m/s^2$}.)$$

Accepted Answer

$$F_{biceps} = \frac{(m$$_{book}$$g \cdot r_{book}) + (m$$_{arm}$$g \cdot r_{arm})}{r_{biceps}} = \frac{(4.0 	imes 9.8 	imes 0.38) + (2.5 	imes 9.8 	imes 0.16)}{0.04} = 456\ \mathrm{N}$$

Question 2

What is the ratio of the biceps force to the combined weight of the arm and the book in the scenario above?

Accepted Answer

$$	ext{Ratio} = \frac{F_{biceps}}{(m_{book} + m_{arm})g} = \frac{456}{63.7} \approx 7.2$$

Question 3

A cyclist applies a force to the pedal, generating a counter-clockwise torque of $$140\ \mathrm{Nm}$$ when the crank is at $$90^\circ to $$the vertical. What is the torque generated when the crank is at $$75^\circ to $$the vertical?

Accepted Answer

$$	au = 140 \sin(75^\circ) = 135.3\ \mathrm{Nm}$$

Question 4

For the same cyclist, what is the torque generated when the crank is at $$180^\circ to $$the vertical?

Accepted Answer

$$	au = 140 \sin(180^\circ) = 0\ \mathrm{Nm}$$

Question 5

A projectile is launched with an initial speed of $$10\ \mathrm{m/s} at an $$angle of $$30^\circ$$ above the horizontal. What is the initial vertical component of its velocity?

Accepted Answer

$$v_{0y} = 10 \sin(30^\circ) = 5\ \mathrm{m/s}$$

Question 6

A projectile is launched from the ground with an initial speed of $$10\ \mathrm{m/s} at$$ $$30^\circ. $$What is the maximum height reached by the projectile? (Use $$g = 9.8\ \mathrm{m/s^2$}.)$$

Accepted Answer

$$h_{max} = \frac{v_{0y}^2}{2g} = \frac{(10 \sin(30^\$$circ))^2$$}{2 	imes 9.8} = \frac{25}{19.6} = 1.28\ \mathrm{m}$$

Question 7

A mass $$m is $$attached to a spring with spring constant $$k$$ and oscillates with amplitude $$A. $$What is the total mechanical energy of the system?

Accepted Answer

$$E = \frac{1}{2}kA^2$

Question 8

A block attached to a spring oscillates with angular frequency $$\omega = 2\pi f. If $$the period of oscillation is $$T$$, what is the relationship between $$\omega$$ and $$T$$?

Accepted Answer

$$\omega = \frac{2\pi}{T}$$

Question 9

A mass on a spring is at its maximum displacement $$A at$$ $$t = 0. $$After a quarter period, where is the mass?

Accepted Answer

At the equilibrium position ($$x = 0)$$

Question 10

A projectile is launched horizontally from a cliff with speed $$v_0. $$Which of the following statements is correct about its horizontal velocity?

Accepted Answer

The horizontal velocity remains constant throughout the motion.

Question 11

A force $$F is $$applied at a distance $$r$$ from the axis of rotation, making an angle $$	heta$$ with the lever arm. What is the torque produced?

Accepted Answer

$$	au = rF \sin	heta$$

Question 12

A uniform rod of length $$L$$ and mass $$m is $$pivoted at one end. What is its moment of inertia about the pivot?

Accepted Answer

$$I = \frac{1}{3}mL^2$

Question 13

A mass $$m is $$attached to a spring with spring constant $$k. $$What is the angular frequency of oscillation?

Accepted Answer

$$\omega = \sqrt{\frac{k}{m}}$$

Question 14

A projectile is launched at an angle $$	heta$$ with initial speed $$v_0. $$Which equation gives the vertical position $$y as a $$function of time $$t$$?

Accepted Answer

$$y = y_0$ + v$$_{0y}$$t - \frac{1}{2}gt^2$

Physics Final Exam Study Guide: Torque and Equilibrium in the Human Arm

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