Q: A disk of mass $$M$$ and radius $$r$$ rolls off a track and becomes a projectile. Neglecting air resistance, what is the maximum height $$H$$ reached above the end of the track if its speed at launch is $$v$$?

$$H = \frac{v^2$ \sin^2$ \theta}{2g}$$

Q: A projectile is launched at an angle $$\theta$$ with initial speed $$v_0. $$What is the horizontal distance traveled before hitting the ground?

$$R = \frac{v_0$^2 \sin 2\theta}{g}$$

Question 1

A satellite of mass $$m$$ moves in an elliptical orbit around the Earth. At point $$C$$, the satellite is at its closest approach to Earth (perigee). If the total mechanical energy of the satellite is negative, what does this imply about the satellite's motion?

Accepted Answer

The satellite is in a bound orbit and will continue to orbit the Earth.

Question 2

A standing wave is formed in a tube of length $$L. $$The pressure variation graph shows a node at $$x=0$$ and an antinode at $$x=L. $$What type of tube is this?

Accepted Answer

A tube open at one end and closed at the other.

Question 3

A torsion balance is used to measure the gravitational constant $$G. $$Two small masses $$m$$ are attached to the ends of a rod of length $$L$$ and suspended by a wire. When two large masses $$M$$ are brought near, the rod rotates due to gravitational attraction. Which equation best represents the equilibrium of torques in the horizontal plane?

Accepted Answer

$$2GMm/L^2 = K$$	heta$$

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 $$v_1$$ and at point 2 is $$v_2$$, which equation expresses conservation of mass?

Accepted Answer

$$A_1v_1 = A_2v_2$$

Question 5

A block of mass $$m$$ attached to a spring of stiffness $$k$$ oscillates with angular frequency $$\omega. If $$another block of mass $$m$$ sticks to it during motion, what is the new angular frequency?

Accepted Answer

$$\omega' = \sqrt{\frac{k}{2m}}$$

Question 6

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

Accepted Answer

$$v = \sqrt{\frac{4gh}{3}}$$

Question 7

A water cannon of mass $$M$$ ejects water at a constant rate $$\mu$$ 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?

Accepted Answer

$$v(t) = -\frac{\mu v_0$}{M}t$$

Question 8

A tube of length $$L is $$open at both ends. What is the fundamental frequency of the standing wave in the tube if the speed of sound is $$v$$?

Accepted Answer

$$f_1$$ = \frac{v}{2L}$$

Question 9

A block of mass $$m$$ attached to a spring oscillates with amplitude $$A. $$What is the total mechanical energy of the oscillator?

Accepted Answer

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

Question 10

A disk of mass $$M$$ and radius $$r$$ rolls off a track and becomes a projectile. Neglecting air resistance, what is the maximum height $$H$$ reached above the end of the track if its speed at launch is $$v$$?

Accepted Answer

$$H = \frac{v^2$ \sin^2$ 	heta}{2g}$$

Question 11

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

Accepted Answer

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

Question 12

A block of mass $$m is $$released from rest at $$x_0 on a $$spring with stiffness $$k. $$What is the equation for its position as a function of time?

Accepted Answer

$$x(t) = x_0$ \cos(\omega t)$$

Question 13

A solid sphere of mass $$M$$ and radius $$R$$ rolls without slipping down an incline of height $$h. $$What is its speed at the bottom?

Accepted Answer

$$v = \sqrt{\frac{10gh}{7}}$$

Question 14

A tube closed at one end and open at the other has a length $$L. $$What is the fundamental frequency of the standing wave in the tube if the speed of sound is $$v$$?

Accepted Answer

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

Question 15

A block of mass $$m is $$attached to a spring and oscillates with angular frequency $$\omega. If $$the spring constant is doubled, what happens to the angular frequency?

Accepted Answer

$$\omega' = \sqrt{2}\omega$$

Question 16

A projectile is launched at an angle $$	heta$$ with initial speed $$v_0. $$What is the horizontal distance traveled before hitting the ground?

Accepted Answer

$$R = \frac{v_0$^2 \sin 2	heta}{g}$$

Question 17

A torsion balance oscillates with period $$T. If $$the torsional constant of the wire is $$K$$ and the moment of inertia is $$I$$, what is the relationship between $$K$$, $$I$$, and $$T$$?

Accepted Answer

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

Question 18

A fluid flows through a horizontal pipe. If the pressure at point 1 is $$P_1$$, the pressure at point 2 is $$P_2$$, and the velocities are $$v_1$$ and $$v_2$$, which equation expresses Bernoulli's principle?

Accepted Answer

$$P_1$$ + \frac{1}{2}\rho $$v_1^2$$ = $$P_2$$ + \frac{1}{2}\rho $$v_2^2$

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

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