Consider a semi infinite circuit like shown above. Magnitude of impedance of inductor and capacitor equal to R (i.e. ωL=1/ωC=R). Find the total impedance of the circuit!
Click here to read the solution.
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Showing posts with label **. Show all posts
A scooter of length l has two wheels. The scooter moves with CM's speed v on a rough plane and is going to turn its direction without change its speed. To turn its direction, the front wheels is rotated to an angle θ.
Calculate the minimum coefficient of friction between the plane and the wheels, μ, so that the scooter does not slip to the plane! Assume CM of the scooter is at the middle.
Click here to read the solution.
A uniform rod of mass M and length l is placed on a rough floor and a rough wall. The rod makes an angle θ to the floor.
What is the minimum coefficient of friction of the floor and the wall in order to prevent the rod from slipping? Assume the floor and the wall made by the same material.
Click here to read the solution.
A sprayer consists of four bent pipes of length 2a for each pipe. The pipes can freely rotate about point O without friction.
An ideal fluid of mass density ρ flows into the sprayer through point O from a large tank of pressure P (assume it's much larger than atmospheric pressure).
After a long time, the sprayer reaches a constant angular velocity, ω. Ignore gravity, energy lost, and turbulence of fluid.
Calculate ω!
Click here to read the solution.
A track consists of two rough planes forming an angle 2θ to each other. It's symmetric to vertical axis and makes an angle φ to the horizontal.
Coefficient of friction of the planes are different. The left side is μ0 and the right side is μ.
Then a ball of radius R and mass M (moment inertia I=ηMR^2) is placed on the track.
As the ball moves, it slips with left side of the track and does not slip with right side of the track.
Find the minimum value of μ! (in μ0, φ, and θ)
Click here to read the solution.
A semicircle arc is charged with charge density distribution at a point, , where λ is total charge in a unit length and α is angle between the diameter and its position.
Calculate the ratio of electrical potential at point A and point O!
Click here to read the solution.
Consider two rods forming V-shape with angle θ and a ball, of radius r, mass m, and moment inertia I, on it. See picture below.
The ball is released at position l from rods' adjacent and start moving because of gravitational acceleration, g. The rods are rough enough, so it prevents the ball from slipping in its movement.
Calculate the initial angular acceleration of the ball!
Click here to read the solution.
Two small balls of mass m are hung by strings of length l to a static nail. The balls are connected with a massless rod of same length, l. Initially, the balls are held at position like shown below.The balls are then released and start moving down.
Calculate the maximum tension, T, of a string during the motion!
Click here to read the solution.
A pencil can be modeled as a hexagonal prism (in this case). When a pencil rolls on a rough floor, it will rotate around one of its edge, and then the next edge will hit the floor, and then the pencil will continue rotating with the edge as a new rotation axis. To simplify the problem, assume mass of the pencil is concentrated at its center.
Calculate the minimum value of coefficient of friction between the pencil and the floor so that the pencil doesn't slip when its edge hits the floor!
Assume the contact between the pencil and floor only occurs at the pencil's edge.
Click here to read the solution.
A uniform rod of length l and mass m is hung by two strings of length l at its two ends, like picture below. Then the rod is rotated to an angle θ and released without any initial velocity or angular velocity.
Calculate the magnitude of acceleration, a, experienced at its end (point A) immediately after it's released!
Click here to read the solution.
A cylinder of radius R and mass M is on a rough inclined plane with inclined angle θ from horizontal. Let μ be the coefficient of friction between the inclined plane and the cylinder. The cylinder makes an angle φ with the generatrix (i.e. the line of intersection of the plane and the horizontal surface). Gravity acceleration is g.
Determine minimum value of μ in order to prevent the cylinder from slipping!
A solid ball with radius r and mass m is located inside an opened wide tank filled with water. The ball has the same density with water.
Calculate work required to get the ball out from the water!
Ignore viscosity and surface tension. Gravity acceleration is g.
(Don't use straight-forward method, this problem doesn't need integration at all)
Two parallel-plate capacitors, A and B, have same area, but separation of plates in capacitor B is a half of A's. The capacitors are in parallel and connected to a voltage source with emf V. Then you insert capacitor B into capacitor A, like shown below.
Calculate the work done by you!
Assume capacitance of capacitor A is C and B is 2C. Ignore edge effect of capacitors.
(If you want a more challenging problem, assume the capacitor is in series, not in parallel)
Consider a thin cylinder's shell with radius r and mass m (inertia mr^2). Two sides of the surface have different characteristics. Outer surface of the shell is a conductor with resistivity ρ and thickness δ (δ<<r). And the inner surface is an insulator which is charged uniformly with surface charge density σ.
A torque τ works on the cylinder. Find the angular acceleration α of the cylinder when its angular velocity is ω!
Assume that magnetic field inside the cylinder is uniform and equals to magnetic field at the middle.
A very thin lens has diameter D and focal length f. It has a weird equation:
(1/f)^n = (1/s)^n + (1/s')^n
where s is an object's position and s' is the image position measured from the lens, and n is an integer where n > 0 (a common lens has n = 1).
Then ray of light come to the lens with a small angle θ from axis of the lens (θ<<1). And a screen is placed behind the lens at focus.
Determine radius of light appear on the screen!
Use (1+x)^n = 1 + nx + 1/2*n(n-1)x^2 for x << 1.
A long solenoid with self-inductance L and radius r is connected to an AC voltage source with frequency ω and max voltage V. Then a thin cylindrical shell with the same length and the same cross-section area with the solenoid is inserted into middle of the solenoid without touching it. It has thickness δ (δ<<r) and resistivity ρ.
Ignore resistance of the solenoid. No current between the solenoid and the cylinder.
Proof that this system is equal to a system where the solenoid is in series with a resistor! Determine the resistance R of this resistor! Assume the system is in vacuum.
Also assume that magnetic field inside the solenoid and the cylinder is uniform and equals to magnetic field at the middle.
A solid hemisphere of radius R is on top of a static ball with the same radius. Then the hemisphere is moved slightly by angle φ (see picture below).
Assume that the hemisphere is always in contact with the ball and also assume that the ball never moves. Determine the initial angular acceleration α of the hemisphere! (gravitational acceleration is g)
Consider a movable prism of inclined angle θ and mass M. The prism is initially at rest on a frictionless horizontal surface.
A particle of mass m falls toward the prism, and then hits the prism. After the collision, the prism will move and the particle's speed forms an angle φ with the normal of prism's surface, like shown below.
Assume that there's no energy lost during the collision.
a) Find the relation between M, m, θ, and φ!
b) Calculate φ if M=5 kg, m=10 kg, and θ=30 deg!
Consider two mirrors and a light source. The mirrors form an angle φ and -φ with a horizontal axis, and the light source is located at horizontal axis and near junction of the two mirrors (see the black dot below).
The light source emits light of total power P with uniform distribution. Assume the mirrors are perfect reflectors. Also assume that this system is only 2 dimensional. Find the distribution of power, dP/dθ, as a function of θ (the angle formed by light and horizontal axis after the light is out from the system)!
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