Q: A disk rotates with a constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$, what is its angular displacement $$\theta$$ after time t$?

$$\theta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$

Q: A uniform beam of length $$L$$ and mass $$m is $$supported horizontally by a hinge at one end and a cable at the other end making an angle $$\theta$$ with the beam. If a mass $$M is $$suspended from the end, what is the tension $$T in $$the cable?

$$T = \frac{(mL/2 + ML)g}{L \sin \theta}$$

Q: A 70-kg ancient statue with a volume of $$3.0 \times 10^4$\ \mathrm{cm^3$}$$ lies at the bottom of the sea. What is the minimum force required to lift it, assuming $$\rho_{\text{water}} = 1000\ \mathrm{kg/m^3$}$$ and $$g = 9.8\ \mathrm{m/s^2$}$$?

$$F = (mg - \rho_{\text{water}} V g)$$

Q: A uniform ladder of mass $$m$$ and length $$L$$ leans against a frictionless wall, making an angle $$\theta$$ with the ground. What is the minimum coefficient of static friction $$\mu_s$$ required at the ground to prevent slipping?

$$\mu_s = \frac{1}{2 \tan \theta}$$

Question 1

A solid cylinder of mass $$M$$ and radius $$R is $$rolling without slipping down an incline of height $$H. $$What is the speed of the center of mass at the bottom of the incline?

Accepted Answer

$$v = \sqrt{\frac{2gH}{1 + \frac{1}{2}}}$$

Question 2

A disk rotates with a constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$, what is its angular displacement $$	heta$$ after time t$?

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$

Question 3

A 60-kg person stands at the edge of a 6.0-m-diameter circular platform (moment of inertia $$1800\ \mathrm{kg\cdot m^2$}) at $$rest. The person runs at $$4.2\ \mathrm{m/s}$$ around the edge. What is the angular velocity of the platform after the person starts running?

Accepted Answer

$$\omega = -0.14\ \mathrm{rad/s}$$

Question 4

A uniform beam of length $$L$$ and mass $$m is $$supported horizontally by a hinge at one end and a cable at the other end making an angle $$	heta$$ with the beam. If a mass $$M is $$suspended from the end, what is the tension $$T in $$the cable?

Accepted Answer

$$T = \frac{(mL/2 + ML)g}{L \sin 	heta}$$

Question 5

A steel wire of length $$L = 1.60\ \mathrm{m}$$ and diameter $$d = 0.20\ \mathrm{cm}$$ stretches by $$\Delta L = 0.25\ \mathrm{cm}$$ under tension. If Young’s modulus for steel is $$E = 2.0 	imes 10$$^{11}$$\ \mathrm{N/m^2$}$$, what is the tension in the wire?

Accepted Answer

$$F = E \frac{\Delta L}{L} A$$

Question 6

A crown of mass $$14.7\ \mathrm{kg}$$ appears to have a mass of $$13.4\ \mathrm{kg}$$ when submerged in water. What is the density of the crown, and is it made of gold ($$ho_{	ext{gold}} = 19,300\ \mathrm{kg/m^3$}$$)?

Accepted Answer

Density is $$\approx 11,300\ \mathrm{kg/m^3$}$$; not gold

Question 7

A hydraulic lift has a small piston of area $$A_1 = 0.01$$\ \mathrm{$$m^2$$}$$ and a large piston of area A_2$ = 1.0\ \mathrm{m^2$}. If a $$force of $$F_1 = 100$$\ \mathrm{N}$$ is applied to the small piston, what is the maximum weight that can be lifted by the large piston (neglecting the weight of the pistons)?

Accepted Answer

$$F_2 = 10$$,000\ \mathrm{N}$$

Question 8

A ball of mass $$m$$ and radius $$r is $$spinning with angular velocity $$\omega$$ and moving with velocity $$v on a $$horizontal surface. If it is rolling without slipping, what is the relationship between $$v$$ and $$\omega$$?

Accepted Answer

$$v = \omega r$$

Question 9

A fluid flows through a pipe with cross-sectional area $$A_1 at $$speed $$v_1$$ and then through a narrower section with area $$A_2. $$What is the speed $$v_2 in $$the narrower section, assuming incompressible flow?

Accepted Answer

$$v_2$$ = $$v_1$$ \frac{$$A_1$$}{$$A_2$$}$$

Question 10

A spinning figure skater pulls in her arms, decreasing her moment of inertia to half its original value. If her initial angular velocity was $$\$$omega_0$$, what is her final angular velocity?

Accepted Answer

$$2 \$$omega_0$$

Question 11

A 70-kg ancient statue with a volume of $$3.0 	imes 10^4$\ \mathrm{cm^3$}$$ lies at the bottom of the sea. What is the minimum force required to lift it, assuming $$ho_{	ext{water}} = 1000\ \mathrm{kg/m^3$}$$ and $$g = 9.8\ \mathrm{m/s^2$}$$?

Accepted Answer

$$F = (mg - ho_{	ext{water}} V g)$$

Question 12

A uniform ladder of mass $$m$$ and length $$L$$ leans against a frictionless wall, making an angle $$	heta$$ with the ground. What is the minimum coefficient of static friction $$\mu_s$$ required at the ground to prevent slipping?

Accepted Answer

$$\mu_s = \frac{1}{2 	an 	heta}$$

Question 13

A fluid is moving through a horizontal pipe that narrows from area $$A_1 to$$ $$A_2. If $$the pressure in the wide section is $$P_1$$ and the speed is $$v_1$$, what is the pressure $$P_2 in $$the narrow section according to Bernoulli’s equation?

Accepted Answer

$$P_2$$ = $$P_1$$ - \frac{1}{2} \rho ($$v_2^2$$ - $$v_1^2$$)$$

Question 14

A torque $$	au is $$applied to a solid disk of moment of inertia $$I. $$What is the resulting angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = \frac{	au}{I}$$

Question 15

A ball is thrown horizontally with velocity $$v$$ from a height $$h$$ above the ground. Neglecting air resistance, what is the angular momentum of the ball about the point on the ground directly below the launch point, just before it hits the ground?

Accepted Answer

$$L = m v \sqrt{2 h g}$$

Question 16

A bar of length $$L is $$used as a lever to lift a rock. If the fulcrum is placed closer to the rock, what happens to the force required to lift the rock?

Accepted Answer

The required force decreases

Question 17

A fluid with viscosity $$\eta$$ flows through a tube of radius $$r$$ and length $$L$$ under a pressure difference $$\Delta P. $$What is the volume flow rate $$Q$$ according to Poiseuille’s law?

Accepted Answer

$$Q = \frac{\pi r^4$ \Delta P}{8 \eta L}$$

Rotational Motion, Angular Momentum, Statics, and Fluids: Study Notes for Physics for Scientists & Engineers (Giancoli, Chapters 10–13)

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