Q: A projectile is launched at an angle $$\theta$$ with initial speed $$v_0. $$What is the maximum height $$H$$ reached by the projectile?

$$H = \dfrac{v_0$^2 \sin^2$ \theta}{2g}$$

Question 1

A satellite of mass $$m is in an $$elliptical orbit around the Earth, with its closest approach (perigee) at distance $$r_p$$ and farthest point (apogee) at distance $$r_a$$ from the center of the Earth. If the speed at perigee is $$v_p$$, what is the speed at apogee $$v_a$$? (Assume no external torques and neglect air resistance.)

Accepted Answer

$$v_a = v_p \dfrac{r_p}{r_a}$$

Question 2

A standing wave is formed in a tube of length $$L. $$The pressure node pattern shows nodes at both ends. Which of the following statements is correct about the harmonic and the frequency of this standing wave?

Accepted Answer

The tube is open at both ends; all harmonics are present and the fundamental frequency is $$f_1$$ = \dfrac{v}{2L}$$.

Question 3

A torsion balance is used to measure the gravitational constant $$G. If $$the torsion constant of the wire is $$K$$ and the moment of inertia of the system is $$I$$, what is the period $$T of $$oscillation of the balance?

Accepted Answer

$$T = 2\pi \sqrt{\dfrac{I}{K}}$$

Question 4

A non-viscous, incompressible fluid flows through a horizontal tube that narrows from area $$A_1 to$$ $$A_2. If $$the velocity at point 1 is $$v_1$$, what is the velocity at point 2, $$v_2$$?

Accepted Answer

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

Question 5

A block of mass $$m is $$attached to an ideal spring of stiffness $$k$$ and oscillates on a frictionless surface. What is the angular frequency $$\omega of $$the oscillation?

Accepted Answer

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

Question 6

A solid disk of mass $$M$$ and radius $$r$$ rolls without slipping down a track of height $$h. $$What is the linear speed $$v of $$the disk at the bottom of the track?

Accepted Answer

$$v = \sqrt{\dfrac{2gh}{1 + \dfrac{1}{2}}}$$

Question 7

A water cannon of mass $$M$$ ejects water at a constant rate $$\alpha$$ with velocity $$v_0$$ relative to the cannon. What is the velocity of the cannon as a function of time, assuming it starts from rest and neglecting friction?

Accepted Answer

$$v(t) = v_0$ \ln\left(\dfrac{M}{M - \alpha t}\right)$$

Question 8

A block of mass $$m is $$released from rest at $$x = A on a $$spring-mass system. What is the equation for its position as a function of time?

Accepted Answer

$$x(t) = A \cos(\omega t)$$

Question 9

A projectile is launched horizontally from a height $$h$$ with initial speed $$v_0. $$What is the time $$t it $$takes to hit the ground?

Accepted Answer

$$t = \sqrt{\dfrac{2h}{g}}$$

Question 10

A manometer is used to measure the pressure difference between two points in a fluid. If the height difference in the manometer is $$h$$ and the density of the manometer fluid is $$\rho$$, what is the pressure difference $$\Delta P$$?

Accepted Answer

$$\Delta P = \rho g h$$

Question 11

A solid sphere and a solid cylinder, both of mass $$M$$ and radius $$R$$, roll without slipping down an incline of height $$h. $$Which one reaches the bottom first and why?

Accepted Answer

The solid sphere, because it has a smaller moment of inertia relative to its mass.

Question 12

A block of mass $$m$$ slides down a frictionless incline of height $$h$$ and then compresses a spring of constant $$k. $$What is the maximum compression $$x_{max} of $$the spring?

Accepted Answer

$$x_{max} = \sqrt{\dfrac{2 m g h}{k}}$$

Question 13

A tube of length $$L is $$open at one end and closed at the other. What is the fundamental frequency $$f_1 of $$the standing wave in the tube?

Accepted Answer

$$f_1$$ = \dfrac{v}{4L}$$

Question 14

A block of mass $$m$$ moving at speed $$v$$ collides and sticks to a stationary block of mass $$2m on a $$frictionless surface. What is the final speed $$v_f of $$the combined mass?

Accepted Answer

$$v_f = \dfrac{v}{3}$$

Question 15

A disk of mass $$M$$ and radius $$R is $$rotating with angular velocity $$\omega. $$What is its rotational kinetic energy?

Accepted Answer

$$K = \dfrac{1}{2} I \$$omega^2$$ = \dfrac{1}{4} M R^2$ \$$omega^2$$

Question 16

A block attached to a spring oscillates with amplitude $$A$$ and angular frequency $$\omega. $$What is the total mechanical energy $$E of $$the oscillator?

Accepted Answer

$$E = \dfrac{1}{2} k A^2$

Question 17

A projectile is launched at an angle $$	heta$$ with initial speed $$v_0. $$What is the maximum height $$H$$ reached by the projectile?

Accepted Answer

$$H = \dfrac{v_0$^2 \sin^2$ 	heta}{2g}$$

Question 18

A block of mass $$m is $$attached to a spring and oscillates with period $$T. $$What is the spring constant $$k$$?

Accepted Answer

$$k = \dfrac{4\pi^2$ m}{T^2$}$$

Question 19

A solid sphere of mass $$M$$ and radius $$R is $$rolling without slipping. What is its moment of inertia about its center?

Accepted Answer

$$I = \dfrac{2}{5} M R^2$

Physics 7A Final Exam Study Guide: Classical Mechanics, Fluids, and Oscillations

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