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A convex mirror has a radius of curvature of 48 cm. How far from the mirror does an object have to be placed to form an image whose magnification is 1/3?
- A convex mirror has a negative focal length given by f = − r / 2, where r is the radius of curvature. f is therefore −24 cm. The magnification is given by m = − i / o where i is the distance from the image to the mirror and o is the distance from the object to the mirror. Since the magnification is 1/3, 1/3 = − i / o, soi = − o / 3. Plugging this into the mirror equation (1 /o + 1 / i = 1 / f) yields 1 / o + 1 / (− o / 3) = 1 / (−24). Simplifying, − 2 / o = 1 / (−24) or o = 48 cm
A test charge 2 nC and mass 10–8 kg is released from rest at point A which has an electric potential of –100 V and travels to point B which has an electric potential of –500 V. What is the speed of the test charge when it reaches point B?
- The change in potential energy can be calculated using ΔPE = qV = (2 × 10–9)(–500 – –100) = –8 × 10–7. Since this represents a loss in potential energy, there must be a gain in kinetic energy of the same amount. Both the initial speed and kinetic energy are zero since the charge started at rest, so the entire change in kinetic energy is due to the new speed. Therefore KE = (1/2) mv2 and v2 = 2KE / m = 2(8 × 10–7) / 10–8 = 160 and v = √160 ≈ 13 m/s. Note that the exact square root is not needed since only choice A is close.
A guitar string is playing notes that are flat (frequency too low). How can the string be made to play the correct notes?
- Increase the tension in the string to increase the wave speed.
- Intuitively, it makes sense that increasing the tension would cause the string to play higher notes, eliminating choices C and D. Specifically, increasing the tension increases the speed of waves traveling along the string: where T is tension and m/ L is the linear mass density of the string. Since the length of the string allowed to vibrate is the same, all harmonic wavelengths will be the same (since λn = 2L / n). This eliminates choice B.
A high current power line is supported 10 m above ground and carries a current from West to East. A negatively charged balloon is released directly below the power line and begins to float straight up toward it. What is the direction of the magnetic force on the balloon due to the magnetic field surrounding the power line?
- Find the direction of the magnetic field created by the wire using the right hand rule. The magnetic field under the wire points North. Once the magnetic field direction is found, find the direction of the magnetic force on the balloon using the left hand rule (since the balloon is negatively charged). The direction of the force on the balloon is to the East
- F1 sinθ – F2 cosθ
- The net force in the horizontal direction is composed only of the horizontal components of F1and F2. The horizontal component of F1 is opposite the angle θ, making it F1 sinθ. Because the angle between F1 and F2 is 90°, the angle between F2and the vertical is 90° – θ, which means the angle between F2 and the horizontal is θ (since the vertical and horizontal lines are perpendicular). Thus the horizontal component of F2 is F2 cosθ. Finally, the net force in the horizontal direction is the sum of those two horizontal components. Plugging in the forces (remember the direction) yields Fnet = F1 sinθ– F2 cosθ.
A car is being lifted by a hydraulically powered jack. The area attached to the car is 100 cm2, while the area where the force is being input is 20 cm2. How much force must be applied to lift a 20 kN car?
- 4 kN
- A hydraulic lift operates via Pascal's Law. Hence the proportion of the force over the area is equal for both sides of the jack. In other words F2 = A2 / A1 ×F1, or solving for the applied force F1 = A1 / A2 × F2. Plugging in the numbers yields 20 cm2 / 100 cm2 × 20 kN = (1/5) × 20 kN = 4 kN.
In a collision between objects (such as cars or tennis balls), which of the following must remain constant between the instant before the collision and the instant after?
1. The total mass of the system?
2. The kinetic energy of the system?
3. The momentum of the system?
- I and III only
- The mass of a system is unaffected by a collision (even if the collision causes the objects to shatter, the sum of the pieces will retain the same total mass). Thus I is true, eliminating choice C. Collisions generally conserve momentum, so III is true, eliminating choice A. Kinetic energy is conserved only in the special case of elastic collisions, where the objects colliding bounce instantly off each other without deforming or otherwise dissipating mechanical energy in other forms. Thus II is false, eliminating choice D.
An eye that is nearsighted:
- needs a diverging lens because the focal length is too short.
- If the eye’s focal length is too short, it would make sense that it would need a diverging lens to cause the light rays to converge at a distance farther away from the lens of the eye. Similarly, if the eye’s focal length is too long, it needs a converging lens to cause the light rays to converge a distance closer to the lens of the eye. This eliminates choices A and D. In particular, nearsightedness or myopia is when the focal length is too short, and therefore correction requires a diverging lens.
A ray of light traveling in Medium #1 (n1 = 1.4) strikes the boundary with Medium #2 at an angle of 60° from the normal. If the angle between the reflected and refracted rays is 75°, then what is the index of refraction of Medium #2?
Torque has which of the following properties?1. It is measured in joules.
2. It is a vector.
3. It can be zero when force is nonzero.
- II and III
- I is false: torque is measured in Newton-meters (N•m). While dimensionally equivalent to joules, they are not considered the same since joules measure scalar quantities such as work and energy. This eliminates choices A, B and D. II is true: torque is indeed a vector, though its vector properties are not typically tested on the MCAT. This eliminates choice A. III is also true: an object can have a force acting on it without exerting a torque if that force's line of action passes through the center of mass of the object (you can think of this as the case in which the angle between the radial vector from the pivot point to where the force is applied and the force vector is 0° or 180°, so sine = 0). This eliminates choices A and B.
A book sits at rest on a horizontal table. It is accelerated with a pushing force. Then, after it has moved a distance s, it is released. Soon after it comes to rest. From the beginning to the end of this problem, what is the net work done on the book?
- 0 J
- The net work done on the book as it moves across a horizontal surface (over which there is no change in gravitational potential energy) must equal the change in kinetic energy. Because the book started and ended at rest, ∆KE = 0 = Wnet.
Saline is flowing at a constant rate through a syringe with a starting diameter of 2 mm and an ending diameter of 4 μm. How many times faster is the saline moving at the end of the syringe than at the start?
Two copper wires are tested for resistance. The first wire is longer and thinner than the second wire. Which wire has the higher resistivity?
- Neither wire, because resistivity depends only on the material, and both wires are made of copper.
- Resistivity is a measure of a material's intrinsic resistance, and is dependent only on the material itself. Since both wires are made of copper, the resistivity of each wire will be the same. Note that answer choice A would be correct if the question were asking for resistance. Note that answer choice D correctly states Ohm's law, but is incorrect because resistance is determined by the wire and is fixed for Ohmic materials, regardless of the voltage and current.
An increase in which of the following will increase the pressure experienced by a submerged plate in a closed container?
1. The area of the plate
2. The thickness of the plate
3. The density of the fluid in which the plate is submerged
- III only
- The amount of pressure experienced is equal to the density of the fluid times the depth of the object times the acceleration due to gravity. Hence, increasing the area of the plate has no impact on the pressure experienced. This makes I false and eliminates choices A and C. The thickness of the plate has no impact on the pressure experienced, making II false and eliminating D. III is true, since the greater the density of the fluid, the greater the pressure experienced by the plate. Therefore choice B is the correct answer.
A dipole consists of two charges separated by a distance of 2 m. The charge on the left is 2 nC and the charge on the right is –2 nC. What is the value and direction of the electric field at a point halfway between the two charges?
- 36 N/C to the right
- The strength of the electric field E from a source charge Q at any distance r from the source charge is E = kQ / r2 where k is Coulomb's constant. The principle of superposition states that the electric fields of multiple source charges can be added. The electric field from the left source charge is ELeft Q = (9 × 109)(2 × 10–9) / 12, = 18 N/C. The direction ofELeft Q is away from the positive charge and toward the negative charge (the direction of force a positive test charge would feel), which is to the right. The electric field from the right source charge is ERight Q= (9 × 109)(–2 × 10–9) / 12, = –18 N/C. The direction of ERight Q is away from the positive charge and toward the negative charge (the direction of force a positive test charge would feel), which is still to the right. The total electric field is then 36 N/C to the right.
Which of the following is true about the normal force?
(A)It is always equal and opposite to the weight.
(B)It can cause an object moving along a surface without colliding to change speed.
(C)It is attributable to the binding forces between atoms and molecules in solid matter.
(D)It cannot contribute to the change in direction of a moving object.
- Process of elimination is the best way to approach this question, unless you happen to recognize choice C as being true at first reading. Choice A is false because the normal force is always perpendicular to the plane of contact between two bodies, whether that plane is horizontal or not (a box on a hill experiences a normal force that is less than and not in the opposite direction to the weight). Because normal forces are perpendicular to the direction of motion of an object moving over the contact surface, they do no work on such an object and therefore do not change the kinetic energy or speed (eliminating choice B). A counterexample to choice D is a banked circular track like a racing oval, where the normal force provides some of the centripetal acceleration of the car going around the track. The reason one does not fall through the floor is indeed because the various electromagnetic bonds resist the breaking that would occur if one's foot passed through the ground.
A bicycle tire has a radius of 33 cm. If the bicycle is traveling at 12 m/s, with what approximate frequency are the tires rotating?
- 6 rotations/sec
- The relationship between translational velocity and rotational frequency depends upon how far the bike moves for each complete rotation of the tires, i.e., on the circumference of the tire. Frequency is the reciprocal of the period, which is the time it takes to complete one revolution. Since period = 2πr / v, then frequency is equal to
When a wave passes from a medium in which it travels slowly to one in which it travels faster, all of the following are true EXCEPT:
(A)The amplitude in the faster medium is less than or equal to the amplitude in the slower medium.
(B)The frequency increases when passing into the faster medium.
(C)The frequency remains the same when passing into the faster medium.
(D)The wavelength is longer in the faster medium.
- When any wave passes from one medium to another, its frequency remains the same. This eliminates choice C, since the statement is true. The relationship between speed, frequency and wavelength is given by v = fλ. If v increases and fstays the same, then λ must also increase. This eliminates choice D. Finally, when any wave hits the boundary between two media, some of the wave will reflect back into the original medium. The wave that is transmitted into the faster medium will have less energy, and therefore a smaller amplitude, than the original wave. This eliminates choice A.
When a sound wave passes from air to water: (A)its frequency increases and its wavelength decreases. (B)its frequency stays the same and its wavelength decreases. (C)its frequency decreases and its wavelength increases. (D)its frequency stays the same and its wavelength increases.
- When any wave passes into a new medium, its frequency remains the same. This eliminates choices A and C. Sound travels faster in water than air since it is a more incompressible medium. The relationship between speed (v), frequency (f) and wavelength (λ) for all waves is given by v = fλ. If vincreases and f stays the same, then λ must also increase.
Jose asks Jill to slide the salad bowl (m = 2 kg) across the table. Jill's initial push gives it an initial speed of 4 m/s, and it comes to a halt directly in front of Jose after traveling 2 meters. What was the magnitude of the kinetic frictional force slowing the bowl during its slide?
A basic circuit consists of a 60 V battery and a resistor. The current on the circuit is 12 A. If the circuit is modified so that two more equivalent resistors are added in series, what is the resulting current on one of the resistors?
- 4 A
- Resistors in series can be added to determine equivalent resistance. In this case, the resistance goes from R to 3R. Using Ohm's law, V = IR, if R increases by 3 then I must decrease by 3, so the current becomes 12 / 3 = 4 A. Since the resistors are in series, each will have the same 4 A current. Note that the voltage on the battery is not needed.
Two fish are submerged in a T-shaped closed container. The first fish is 15 centimeters from the bottom, the second is 30 centimeters from the bottom. Compared to the second fish, the first fish experiences what change in gauge pressure? (A)The first fish experiences half the gauge pressure. (B)The first fish experiences twice the gauge pressure. (C)A change in gauge pressure cannot be determined, because gauge pressure depends on the shape of the container. (D)A change in gauge pressure cannot be determined, because gauge pressure depends on the depth of the fish.
- Gauge pressure depends only on the density of the fluid and the depth of the object in that fluid. It does not depend on the distance from the bottom, or the shape of the container. Given the information in the question, it is not possible to determine the gauge pressure, eliminating choices A and B. Since gauge pressure does not depend on the shape of the container, choice C can be eliminated, leaving only choice D.
- Concave mirror
- All real images are inverted and all virtual images are upright. The fact that the image is upright shows it is virtual. For mirrors, virtual images appear on the side opposite the object, whereas for lenses, they appear on the same side as the object. Since the image is on the opposite side, the optical element must be a mirror, eliminating choices C and D. Convex mirrors always produce images that are smaller than the object (from 1 / o + 1 / i = 1 / f, where f is negative, yielding i also negative and smaller than o, and then m = − i / o < 1). Since the image is larger than the object, the mirror must be concave.
A pipe closed at one end supports a standing wave vibrating in its 3rd harmonic mode. If the closed end is then opened, and a standing wave is created with twice the previous frequency, then what is the pipe’s new harmonic mode?
- The harmonic mode, denoted by n, must be an integer. This eliminates choice A. For a pipe closed at one end, the nth harmonic frequency is given byfn = vn / 4L, where v is the speed of sound and L is the length of the pipe. Since the pipe is playing in its 3rd harmonic mode, f = (3v) / (4L). For a pipe open at both ends, fn = vn / 2L. Doubling the original frequency and plugging into this equation yields (2)(3v) / 4L = vn / 2L. Canceling v and L yields 6/4 = n / 2, so n = 3.
When light passes through a certain medium that is surrounded by air, the critical angle for total internal reflection is 45°. If that medium is submerged in water (n = 1.33), what is the new critical angle?
Which of the following is a description of a system in translational equilibrium? (A)A ball rolling around a circular track at constant speed. (B)A car driving at a constant speed up a straight hill with an angle of inclination θ. (C)An object in free fall. (D)Two unequal, free-hanging masses attached by a rope running over a vertical pulley (i.e. an Atwood machine).
- Translational equilibrium means the net force equals zero, so acceleration equals zero, which means that the velocity must be constant. A ball rolling at a constant speed but in a curved path does not have constant velocity, because direction is changing (eliminating choice A). An object in free fall is accelerating at g (eliminating choice C), and an Atwood machine in free motion has an acceleration of (m2 – m1)g / (m1 + m2) (eliminating choice D). Note that calculating the actual acceleration value is unnecessary since the acceleration must be nonzero so long as the masses are unequal and free to move.
A diving bird is chasing a fish. If it dives in the ocean from a depth of 10 m to a depth of 30 m, what happens to the total pressure on the bird? (A)It decreases by a factor of 3 (B)It stays the same (C)It increases by a factor of 3 (D)It increases, but the exact amount cannot be determined from the information given
- This question is asking about total pressure, which is the sum of hydrostatic gauge pressure and the atmospheric pressure. Ptotal = Pgauge + Patm. The bird is swimming deeper, so the pressure must increase, which eliminates A and B. Pgauge =ρfluidgD, meaning that gauge pressure is directly proportional to depth. Since the depth triples, the gauge pressure will triple. However, the question is asking about total pressure, which includes the constant atmospheric pressure. This eliminates answer choice C, meaning that answer choice D must be correct.
Two blocks start at height h: block 1 on a frictionless ramp of constant angle of inclination, block 2 suspended in midair. Both blocks are released at the same moment. How does the average power exerted on the blocks during their descent compare? (A)P1>P2 (B)P1=P2 (C)P1<P2 (D)Cannot be determined based on the information given.
- Power is work / time. The work done by gravity on each block is the same, because the change in height of each block is the same. The normal force on block 1 does no work because it is perpendicular to displacement. Since there is no friction on the plane, the net work done on each block must therefore be the same. The time it takes block 2 to descend over the displacement h is less than the time it takes block 1 to displace a longer distance down the ramp (h / sinθ), both because block 2 undergoes less displacement and because block 2 experiences greater acceleration (g versus g sinθ). Less time for the same work means more power.
A student stands 6 m from a speaker emitting a spherical sound wave. The intensity of the sound at the student’s position is 2 × 10–10W/m2. If the student then steps backward an additional 6 m from the speaker, what is the intensity at the new position?
- 5 × 10–11 W/m2
- For a spherical wave, intensity (I) is inversely proportional to the square of the distance (r) from the source to the detector. The student is increasing her distance to the source which would therefore decrease the intensity. This eliminates choices A and B. Specifically, r doubles. So since I is proportional to 1 / r2, doubling r would cause the intensity to decrease by a factor of 4. (1/4)(2 × 10–10 W/m2) = 5 × 10–11 W/m2.
A certain simple pendulum oscillating with a small angle has the same period as an ideal mass/spring system. If the length of the pendulum is doubled, its period will remain the same as the mass/spring system if: (A)the spring constant is doubled. (B)the pendulum mass is doubled. (C)the mass on the spring is halved. (D)the mass on the spring is doubled.
A circuit consists of a battery and 4 identical resistors in parallel. The current in each resistor is 12 A. If one of the resistors burns out, what will be the current in each of the remaining 3 resistors?
- 12 A
- The battery remains the same and will provide the same voltage both with the 4 resistors and with the 3 resistors. Since the resistors are in parallel, the voltage drop on each must be the same as the voltage provided by the battery, and will be unchanged when one resistor burns out. Since bothV and R are unchanged for each resistor by the burned out resistor, I is also unchanged for each resistor, and the current in each resistor is still 12 A. Note that what does change is the total current on the circuit, from 48 A with 4 resistors (4 × 12 A) to 36 A with 3 resistors (3 × 12 A). Since the resistors are in parallel, fewer resistors will mean a higher equivalent resistance, so the total current is reduced, even though the current on each resistor is unchanged.
A person stands at an equal distance between two speakers which produce identical sound waves. If the person moves so that she is 1.5 m closer to one speaker than the other, she is in a “quiet zone” (i.e. the waves experience destructive interference). The wavelength of the sound waves could be all of the following EXCEPT: (A)0.6 m (B)1 m (C)3 m (D)6 m
- If the path difference from two wave sources to the detector is an integer multiple of wavelengths, then the waves will experience constructive interference (i.e. compressions will line up with compressions, rarefactions will line up with rarefactions). This can be expressed as d2 – d1 =mλ, where d2 and d1 are lengths of the paths traveled by the waves, m is an integer and λ is the wavelength. Similarly, if the path difference is equal to an integer-and-a-half times the wavelength, the waves will experience destructive interference: d2 –d1 = (m + 1/2)λ. Since the waves are experiencing destructive interference, 1.5 = (m + 1/2)λ. So 1.5 could equal λ / 2 or 3λ / 2 or 5λ / 2, etc. Solving forλ, we get that λ = 2(1.5) = 3 or (2/3)(1.5) = 1 or (2/5)(1.5) = 0.6, etc. (all answers in meters). Choice D is the only answer not possible.
A bobsled team of four men, each of mass m,pushes the sled of mass M over a horizontal displacement d before leaping into the sled and beginning to slide down the hill. The total length of the winding track from the top of the hill to the bottom is L, and the height difference between the top of the hill and the bottom is Δh. If the sled starts at rest and we assume the whole track is frictionless, how does the final kinetic energy of the sled at the bottom of the hill (prior to the brakes being applied) compare to the initial potential energy of the sled at the top of the hill? (A) KEf > PEi (B) KEf = PEi (C) KEf < PEi (D) It is not possible to uniquely determine the relationship between these two energies.
- With no friction, this is a case of conservation of mechanical energy after work is done on the combined mass by the running men. The conservation relation can be written KEfinal = PEinitial+ Wexternal or, if you take the speed achieved by the bobsled team just as they leap into the sled, KEfinal= PEinitial + KEinitial. Either way it is clear that the final kinetic energy exceeds the initial potential energy of the sled.
The electric potential at point p due to source charge Q is ϕ. Which of the following calculates the distance from p to Q? Use k for Coulomb's constant. (A) kQ / ϕ2 (B )kQ2 / ϕ (C) kQ / ϕ (D) ϕ / kQ
- The electric potential from a source charge at a point is given by ϕ = kQ / r where r is the distance between the point and the source charge. Solving for r yields r = kQ / ϕ. Without the formula for the electric potential due to a point charge, process of elimination using units would still work (only choice C has the correct units of meters).
A block with a mass of 5 kg rests on a horizontal table (µs = 0.3, µk = 0.2). A constant horizontal pushing force of 20 N is applied. How far does the block move in 2 seconds?
- Like many friction problems, this is an example of a dynamics leading to kinematics problem. Begin by confirming that the block does in fact move, which can be done by determining the maximum static frictional force: Fs max = µsmg = (0.3)(5)(10) = 15 N. That’s less than the applied force, indicating that the block does indeed accelerate across the table, with kinetic friction working against this sliding motion. Next, the net force is needed, which can be found using Newton’s second law. Fx net = Fapplied – Fkinetic friction = Fapplied – µkmg = 20 – (0.2)(5)(10) = 20 – 10 = 10 N. The acceleration can be determined. a =Fx net / m = 10 / 5 = 2 m/s2. Finally, kinematics can be used to find the distance. a = 2 m/s2; v0 = 0 m/s; t= 2 s; d = ? so v is missing and d = v0t + 0.5at2 = 0 + (0.5)(2)(2)2 = 4 m.
Point charge A exerts a force F on point charge B. If the charge on charge B is doubled and the distance between the charges is halved, what is the resulting force exerted by charge A on charge B? (A) F (B) 2F (C) 4F (D) 8F
- The equation to calculate the force between two point charges is F = kQAQB / r2. The result of QBgoing up by a factor of 2 is F goes up by a factor of 2. The result of r going down by a factor of 2 is Fgoes up by a factor of 4 (since r is in the denominator and squared, F is inversely proportional to r squared). The net result is the force increases by a factor of 8. Algebraically, FNew =kQA(2QB) / (0.5r)2 = 8 (kQAQB / r2).
The mechanical advantage of a machine is a ratio of the force required to lift an object a given distance to the force required to move that object the same vertical distance using a simple machine. A ramp is one example of a simple machine that allows one to exert less force in moving an object from a lower to a higher point. Suppose a box is pushed along a 5 meter long ramp with a rise of 2 meters (with negligible friction between the box and the ramp). What mechanical advantage is gained by using the ramp instead of just lifting the box straight up?
- Because the work involved is the same whether the mass is lifted straight up or pushed up the ramp, mechanical advantage can also be calculated as the ratio of the distance over which the force is exerted using the simple machine to the vertical displacement: . Thus in this case MA = 5/2 = 2.5.
The source of a sound emitting a frequency of 1200 Hz moves with one third the speed of sound. If a detector chases the source (i.e. is located behind the sound source and moving in the same direction) at a speed equal to half the speed of sound, then what frequency does the detector measure?
- 1350 Hz
- According to the Doppler Effect, when the source and the detector of a wave move closer together, the detector receives a frequency that is higher than what is emitted by the source. Similarly, when the source and detector move apart, the detector receives a frequency lower than what is emitted. Since the detector is chasing the source (i.e. moving in the same direction) and the detector is moving faster than the source, they are getting closer together. The detector would therefore receive a frequency that is higher than 1200 Hz. This eliminates choices A and B. Mathematically,
- where fS is the frequency emitted by the source, fD is the frequency received by the detector, v is the speed of sound, vDis the speed of the detector, and vS is the speed of the source. Since the detector is moving toward the source, the numerator increases, so choose the + sign. Since the source is moving away from the detector, the denominator increases as well, so choose the + sign again. This yields
An ideal projectile launched from the ground at an angle of 45° up from the horizontal has a total flight time of T. Which of the following correctly expresses a relationship between the displacement d, velocity v, and acceleration a at T / 2 compared to time T (the instant before it hits the ground)?
(A) 2 × |dT/2| > |dT|
(B) |vT/2| = |vT|
(C) |aT/2| > |aT|
(D) vT/2 = 0
- At half the total flight time, a projectile launched upward from even ground upon which it will eventually land will be at the apex of its flight. At this time, all of its velocity is in the horizontal direction (meaning it has slowed down, eliminating choice B, but is still moving, eliminating choice D). Its acceleration throughout the flight is a constant gdownward, eliminating choice C. That only leaves answer choice A, which is correct: The magnitude of the displacement from the launch point to the apex is a diagonal line that is the hypotenuse of a right triangle with the max height of the projectile as one leg and half the range (R = |dT|) as the other. Twice this hypotenuse must therefore be greater than the total range dT.
An electron with charge –e and mass mmoves with velocity v in the positive x direction through an external magnetic field B that points in the negative z direction. Assuming the effects of gravity are negligible, which of the following best describe the the electron's circular path as viewed from the positive zperspective (looking in the negative zdirection)? (A)Counterclockwise circle with radius mv / eB (B)Clockwise circle with radius mv / eB (C)Counterclockwise circle with radius eB / mv (D)Clockwise circle with radius eB / mv
- Since the charge is negative, the left hand rule is used to determine the direction of the magnetic force. The velocity is in the positive x direction and the magnetic field is in the negative z direction, so the initial magnetic force is in the negative ydirection. Thus the acceleration on the electron results in the electron moving in a clockwise circle, eliminating choices A and C. Since the magnetic force is providing the centripetal force, the two forces can be set equal to each other, yielding mv2 /r = evB. Solving for r gives mv / eB.
A 10 kg, 4 m long plank of wood is going to be used as a teeter-totter for a brother and sister. The brother has a mass of 30 kg and the sister a mass of 20 kg. If the brother and sister sit at opposite ends of the plank, how far from the brother should the fulcrum be in order for the teeter-totter to be balanced?
- 1.67 m
- To solve this problem, we use the equation for the center of mass in one dimension, remembering to include a term for the mass of the plank,M, at its geometric center (i.e. the midpoint of the plank). Setting the location of the brother as the origin (x1 = 0), we see that:
When red light (λ = 700 nm in a vacuum) passes into a medium with an index of refraction of n = 1.4, its new wavelength is equal to which of the following?
- 500 nm
- The speed of light, v, in a medium with index of refraction n, is given by v = c / n, where c is the speed of light in a vacuum (i.e. 3 × 108 m/s). For all waves, v = fλ, where f is the frequency and λ is the wavelength. When a wave passes into a new medium, its frequency stays the same, so we can say that, in a vacuum, c = fλvacuum and, in the medium, (c / n) = fλmedium. Dividing the first equation by the second yields n = λvacuum / λmedium, which means that λmedium = λvacuum / n = (700 nm) / 1.4 = 500 nm.
A mass m is released from rest at the top of a frictionless inclined plane with a 3 meter base and a 4 meter height. With what speed does the mass reach the ground at the base of the plane?
- 9 m/s
- The force down an inclined plane due to gravity isFdown plane = mg sinθ, so a = g sinθ, v0 = 0, and d = 5 m (because the ramp is the hypotenuse of a 3-4-5 right triangle). Thus, kinematics can be used to find the final speed (in this case time is missing):
- Note, this can also be solved using conservation of energy.
Seltzer water is flowing through a pipe of uniform diameter for distribution. The pressure is constant throughout the pipe, and the pipe is sloped downward in such a manner that the end for distribution is 10 cm lower than the beginning. What is the water's speed at the end of the pipe if the initial speed is 1 m/s?
- 1.73 m/s
- This question has all the hallmarks of a Bernoulli's equation question: it involves pressure, density, speed, and height. Start from the full Bernoulli equation: P1 + 1/2 ρv12 + ρgy1 = P2 + 1/2ρv22 + ρgy2. Since the pressure is constant, the term can be dropped: 1/2 ρv12 + ρgy1 = 1/2 ρv22 + ρgy2. Now density is in every term and can be cancelled out: 1/2 v12 + gy1 = 1/2 v22 + gy2. Next, the end speed can be solved for: 2 (1/2 v12 + gy1 – gy2) =v22 = v12 + 2g(y1 – y2). Finally, the numbers can be plugged in v22 = (1 m/s)2 + 2 × 10 m/s2 × 0.1 m = 3 m2/s2, so v2 = √3 or ≈ 1.7 m/s.
Which of the following figures represents the paths of red and blue light passing through a piece of glass surrounded by air?
- When light enters a medium with a larger index of refraction (e.g. glass), it will bend toward the normal. Similarly, when light enters a medium with a smaller index of refraction, it will bend away from the normal. This eliminates choice C and D since both bends are toward the normal. Due to dispersion, the shorter the wavelength of light, the slower it travels through glass. The slower it travels, the higher its index of refraction, and thus the more it will bend going into and out of glass (according to Snell's Law). Since blue light has a shorter wavelength than red light, choice A must be correct.
A capacitor is connected in a circuit to a battery and allowed to fully charge. While the capacitor is still connected to the battery, a dielectric is inserted between the plates of the capacitor. If the dielectric constant is K
, all of the following are true EXCEPT:
The charge stored on the capacitor increases by a factor of K.
The potential energy stored on the capacitor increases by a factor of K.
The voltage across the capacitor increases by a factor of K.
The electric field created by the capacitor remains the same
- When the dielectric is inserted, the capacitance of the capacitor increases by a factor of K. Since the battery is still connected, the capacitor will continue charging until the voltage on the capacitor is the same as the voltage on the battery. Therefore V will remain the same, making choice C false and the correct answer. Using Q = CV with V the same and C increasing by a factor of K, then Q must also increase by a factor of K, eliminating choice A. Using PE = (1/2) QV then PE must also increase by a factor of K, eliminating choice B. Using V = Ed with V the same, then E must also be the same, eliminating choice D.
Which of the following kinematic descriptions is physically impossible? (A)An object in motion for some period of time has zero average velocity but positive average speed. (B)An object maintains constant speed but has nonzero acceleration. (C)An object in motion has zero displacement but positive total distance traveled. (D)An object in motion for some period of time has an average velocity with a greater magnitude than its average speed.
- This question can be answered using process of elimination. Choices A and C are both possible (and therefore eliminated as answer choices) because an object traveling in a closed curved path (like a circle) will have zero displacement and therefore zero average velocity but not zero distance or average speed. Choice B is possible, therefore eliminated, because it is the case for uniform circular motion. Hence, choice D must be correct. Average velocity has magnitude of displacement / time, whereas average speed equals total distance / time. The time is the same for both, and displacement ≤ total distance, so average velocity ≤ average speed. Put another way, there’s no shorter path than a straight line, for which the magnitude of displacement equals the total distance. For any other type of path, the displacement is shorter than the distance traveled.
The earth’s orbit around the sun is very nearly a perfect circle. The distance from the earth to the sun is 1.5 × 1011 meters, and one earth year is about π × 107 seconds. What is the magnitude of the earth’s acceleration toward the sun?
- 6 × 10–3 m/s2Because the orbit is circular, the acceleration experienced by the earth is centripetal. Using r as the radius and t as the period yields:
A car of mass m drives around a circular banked track of radius R at constant speed v, and maintains a constant height h. What best describes the work done by the normal force on the car during one quarter of a revolution around the track? (A) WN < 0 (B) WN = 0 (C) WN > 0 (D) The work done by the normal force can be positive or negative depending upon the direction the car is moving.
- Work done by a force on an object depends upon the component of the force along the direction of the displacement of the object. Anything moving in a circular path is, at any given instant, displacing in a direction tangent to the path (the instantaneous velocity), while it is accelerating along the radius toward the center of the circle (centripetal acceleration). Thus the force and displacement are perpendicular, and so the work done by the force is zero. It is also useful to consider that, because the car is not changing height or speed, it is neither gaining nor losing mechanical energy, so no work is being done on it by an outside force.
An alternating current (AC) circuit consists of an alternating voltage source with a maximum voltage of 28 V and a maximum current of 7 A. What is the average power provided by the voltage source?
- 98 W
- Average power for an AC circuit is IrmsVrmswhere Irms = Imax / √2 and Vrms = Vmax / √2. In this case, power = (7 / √2)(28 / √2) = 196 / 2 = 98 W.