Block B with a mass of 5 kg is riding on Cart A with a mass of 7 kg, without slipping. A horizontal force, F, is being applied to Block B so that the combined system is accelerating to the right at 2.5 m/s2. a. What is the minimum static coefficient of friction between Block B and Cart A, required to prevent slipping between the block and the cart? b. What is the magnitude of the force, F, being applied to Block B?
The figure below shows a conducting bar of mass m = 0.200 kg that can slide without friction on a pair of rails separated by a distance I = 1.20 m and located on an inclined plane that makes an angle q = 25.0o with respect to the ground. The resistance of the resistor is R = 1.00 W, and a uniform magnetic field of magnitude B = 0.500 T is directed downward, perpendicular to the ground, over the entire region through which the bar moves. With what constant speed v does the bar slide along the rails?
A car (m = 1760 kg) is parked on a road that rises 16∘ above the horizontal. What are the magnitudes of (a) the normal force and (b) the static frictional force that the ground exerts on the tires? (a) Number Units (b) Number Units
Anza has a mass of 59.1 kg. Ben has a mass of 53.9 kg. The cart has a mass of 26.7 kg. Anza, Ben, and the cart all started from rest (in the laboratory frame of reference). While Anza remained standing on the cart, Ben jumped to the right (just as was the case in the previous question). Ben jumped to the right with a speed of 3.5 m/s relative to the moving cart (so Anza would have seen Ben moving away from her with a speed of 3.5 m/s). Then (after Ben jumped to the right) Anza jumped to the left. Anza jumped with a horizontal speed of 3.3 m/s relative to the moving cart. In other words, after both Ben and Anza jumped, Anza would have seen the cart moving away from her with a speed of 3.3 meters per second. After Anza jumped, what was the velocity of the cart (in meters per second) in the laboratory reference frame? Consider the positive direction to point to the right. The sign of your answer (positive or negative) is important.
Two hard rubber spheres, each of mass m = 15.0 g, are rubbed with fur on a dry day and are then suspended with two insulating strings of length L = 5.00 cm whose support points are a distance d = 3.00 cm from each other as shown in the figure. During the rubbing process, one sphere receives exactly twice the charge of the other. They are observed to hang at equilibrium, each at an angle of θ = 10.0∘ with the vertical. Find the amount of charge on each sphere.
An object of mass m1 = 4.00 kg is tied to an object of mass m2 = 4.50 kg with String 1 of length ℓ = 0.500 m. The combination is swung in a vertical circular path on a second string, String 2, of length ℓ = 0.500 m. During the motion, the two strings are collinear at all times as shown in the figure. At the top of its motion, m2 is traveling at v = 5.00 m/s. (a) What is the tension in String 1 at this instant? N (b) What is the tension in String 2 at this instant? N (c) Which string will break first if the combination is rotated faster and faster? string 1 string 2
In the figure below, capacitor 1(C1 = 30.0 μF) initially has a potential difference of 55.0 V and capacitor 2(C2 = 5.10 μF) has none. The switches are then closed simultaneously. (a) Find the final charge on each capacitor after a long time has passed. Q1 = CQ2 = (b) Calculate the percentage of the initial stored energy that was lost when the switches were closed. %
Three masses (m1 = 2.00 kg, m2 = 1.00 kg, and m3 = 5.00 kg) are connected with inextensible ropes of negligible mass, as shown in the figure. The pulley is massless and frictionless, and the coefficient of kinetic friction between m1 and the incline (θ = 40.0∘) is μ = 0.300. The block of mass m3 moves downward when a force F = 100 N is applied on it. Find: (a) the acceleration of each block, and (b) the tension in each string.
where x = 1.56. As shown in the top-view diagram, a 46.9-N force is applied to the outer edge of a door of width 1.56 m. Find the torque when the force acts perpendicular to the door. N⋅m
A book of mass M is positioned against a vertical wall. The coefficient of friction between the book and the wall is μ. You wish to keep the book from falling by pushing on it with a force F applied at an angle θ with respect to the horizontal (−π/2 < θ < π/2), as shown in the figure. (a) For a given θ, what is the minimum F required? (b) For what θ is this minimum F the smallest? What is the corresponding minimum F? Answers a) Fmin = Mg μscosθ+sinθ b) θ = tan−1 1/μs, Fmin = Mg μs2+1
In the figure two tiny conducting balls of identical mass m and identical charge q hang from nonconducting threads of length L. Assume that θ is so small that tanθ can be replaced by its approximate equal, sinθ. If L = 110 cm, m = 14 g, and x = 4.3 cm, what is the magnitude of q? Number Units
What must be the distance in meters between point charge q1 = 27.1 μC and point charge q2 = −44.1 μC for the electrostatic force between them to have a magnitude of 7.51 N? Number Units
A bucket has a mass of 30 kg when filled with sand. You need lift it to the roof of a building 10 meters tall using a rope that has a mass of 0.3 kg/m. You need 1 meter of rope to secure the rope to the bucket. Once the bucket reaches the top of the building, it has mass 28 kg (not including the rope) because sand leaked out of a hole in the bucket at a constant rate while you were steadily lifting it to the top of the building. The work done lifting the bucket, sand and rope to the top of the building is: Joules (Recall that acceleration due to gravity is 9.81 m/sec2.)
Three particles are fixed on an x axis. Particle 1 of charge q1 is at x = −a and particle 2 of charge q2 is at x = +a. If their net electrostatic force on particle 3 of charge Q is to be zero, what must be the ratio q1/q2 when particle 3 is at (a) x = +0.381a and (b) x = +1.35a? (a) Number Units (b) Number Units
What is the magnitude of the electrostatic force between a singly charged sodium ion (Na+, of charge +e) and an adjacent singly charged chlorine ion (Cl−; of charge −e) in a salt crystal if their separation is 2.854×10−10 m? Number Units
In the figure particle 1 of charge +4e is above a floor by distance d1 = 3.80 mm and particle 2 of charge +7 e is on the floor, at distance d2 = 7.00 mm horizontally from particle 1. What is the x component of the electrostatic force on particle 2 due to particle 1? Number Units
In the figure two tiny conducting balls of identical mass m and identical charge q hang from nonconducting threads of length L. Assume that θ is so small that tanθ can be replaced by its approximate equal, sinθ. If L = 120 cm, m = 8.5 g, and x = 7.4 cm, what is the magnitude of q? Number Units
How many coulombs of positive charge are in 1.28 mol of neutral molecular-hydrogen gas (H2)? Number Units
Three point charges are arranged as shown in the figure below. (Take q1 = 5.70 nC, q2 = 4.60 nC, and q3 = −2.82 nC.) (a) Find the magnitude of the electric force on the particle at the origin. N (b) Find the direction of the electric force on the particle at the origin. (counterclockwise from the +x-axis)
Question 1 Consider the four solid spheres in the figure. In each case the point P is at the same distance from the centre of the sphere. For the situations listed below, rank the spheres according to the magnitude of the electric field at point P, from the largest to the smallest. Indicate any ties. Motivate your answers clearly. (a) (c) (d) (a) The spheres are non-conducting and uniformly charged with the same charge density. (b) The spheres are non-conducting and uniformly charged with the same total charge. Answers: (a) c = d, b, a (b) a = b, c, d
Question 2 Apply what you know about conservation laws to the following questions. (a) Two protons are held fixed on the y-axis at y = a and y = −a. You release an electron (mass me) on the x-axis at x = 2 a. What is the maximum speed the electron will reach? (b) An electron (mass me) and proton (mass mp) are held a distance r apart. They are released simultaneously and begin to accelerate towards each other. What is the electron's speed by the time the distance between the charges has halved? Answers: (a) vmax = [e2 ameπϵ0(1−15)]1 /2 (b) ve = [14πϵ02 e2 mpme(me+mp)r]1 /2 Question 3 (a) Show that the electric field inside a solid charged sphere with a uniform charge density of ρ is E→(r→) = ρr→/(3ϵ0). Here r→ is a position vector relative to the centre of the sphere. Use known results for spherically symmetric charge distributions. (b) Suppose a spherical cavity is hollowed out of the sphere. The centre of the cavity is at position a→ relative to the centre of the sphere. Derive an expression for the electric field inside the cavity. You should find that it is perfectly uniform. Hint: Use your result in (a) together with the superposition principle.
Question 4 A charge +q is suspended a distance d/2 above the centre of a flat square surface with side-length d. (a) Can the electric flux through the square surface be calculated using ΦE = E→⋅A→? Explain your answer. (b) Calculate the electric flux through the square surface using Gauss's law, and explain your reasoning. Question 5 Four identical charges, each with charge q, have been placed on the corners of a square with side-length L. How much work would you need to perform in order to move the charges to the corners of a square with side-length 3L? Answer: Wex = −14πϵ0 q2 L23[4+2]
Question 6 Two small cubes each carry a charge of Q, and are connected by an ideal spring with spring constant k. Initially the spring is at its natural length L, as in figure (a). The boxes move frictionlessly on a horizontal surface, but do experience air-resistance when they are in motion. When the boxes are released from rest they oscillate back and forth, and eventually come to rest at a distance 2 L apart, as in figure (b). During this process, what fraction of the system's b initial mechanical energy was lost to air- resistance? Answer: A fraction of 3/8 was lost.
A disc rotates about a spindle with a constant angular velocity ω = 2.1 rad/s, as shown in the figure. A small block is situated in a frictionless slot at a radius of r0 = 0.21 m, where it is initally at rest with respect to the disc. The outer radius of the disc is R = 1.38 m. If the block is allowed to slide freely one imagines that it will slide outward past the outer edge of the disc. What is the magnitude of the block's velocity as it passes across the outer edge of the disc? Hint: This is just tricky. No other way to put it really! There is NO FORCE in the radial direction, but there will be a "centrifugal effect". Remember that the final velocity will have radial and transverse components!
A small 1.14 kg block slides up a curved inclined surface described by y = k⋅ln(x/10) where k = 20.9 m, and where x and y are both in metres; at a speed of v = 9.6 m/s when x = 20 m, as shown in the figure. The coefficient of kinetic friction is μk = 0.19. What is the radius of curvature of the path, ρ, at this instant? What is the normal force, N, acting on the block from the inclined surface? What is the tangential accleration?
A spacecraft orbits the Earth unpowered in a circular path at a height of 314 km above the Earth's surface, as shown in the figure. At position A thrusters are applied briefly that DOUBLE the satellite's speed. You may assume: RE = 6,380,000 m; ME = 5.98E24 kg and G = 6.673E−11 N.m2/kg2. What is the period the spacecraft before the thrusters are applied? What is the angular velocity of the craft at B with respect to the Earth's centre, ωB, when the radius has increased by 19.1% from the circular orbit radius?