Work and conservation of energy

Work

Problem:

A cyclist intends to cycle up a 8o hill whose vertical height is 150 m with constant speed.  If each complete revolution of the pedals moves the bike 6 m along its path, calculate the average force that must be exerted on the pedals tangent to their circular path.  Neglect work done by friction and other losses.  The pedals turn in a circle of diameter 30 cm.  The total mass of the cyclist and his bike is 100 kg.

Solution:


Conservation of energy

Problem:

A pencil of length L is held vertically with its point on a desk and then allowed to fall over.  Assuming that the point does not slip, find the speed of the eraser just as it strikes the desk.  Compare this with the speed that would result from free fall from a height equal to the length of the pencil.

Solution:

Problem:

Two thin beams of mass m and length l are connected by a frictionless hinge and thread.

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The system rests on a smooth surface.  At t = 0 the thread is cut.  Neglect the mass of the hinge and the treat.
(a)  Find the speed of the hinge when it hits the floor.
(b)  Find the time it takes for the hinge to hit the floor in terms of an integral.

Solution:

Problem:

Galileo wants to demonstrate that unequal masses fall with a uniform acceleration by dropping two spherically symmetric masses simultaneously from the same height z.  Ideally we know that they should hit the ground at the same time, t = (2z/g)½,  where g is the acceleration due to gravity.
Unfortunately for Galileo, the surface of the first sphere is slightly irregular, so the otherwise negligible air drag induces a rotation with an angular frequency proportional to the velocity, ω = εv1.  The second sphere does not rotate.  Assume that ε is constant and ignore possible heating of the air by friction.
Find the ratio of impact times t1/t2 of the two spheres.  The answer should involve only the coefficient ε and the mass m1 and moment of inertia I1 of the first sphere.

Solution:

Problem:

Find the fraction of the kinetic energy that is translational and rotational when
(a)  a hoop
(b)  a disc and
(c)  a sphere rolls down an inclined plane of height h.  Find the velocity at the bottom in each case.  Compare with a block sliding without friction down the plane.

Solution:

Problem:

A sphere of radius R and uniformly distributed mass m is released at rest and begins moving down an inclined plane which makes an angle q with the horizontal.   The sphere reaches a final position as indicated, with its center at ∆z = -h relative to its initial position.  The gravitational force on the sphere is F = -mg k.

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(a)  If the surface is frictionless, so the sphere slips without rotating,
    (i)  what is the time t required to go from the initial to the final position and
    (ii )  what is the final velocity of the sphere?

(b)  Now assume that the surface has sufficient friction so that the sphere rolls without slipping.  In that case, what is the final velocity of the central point of the sphere?

Solution:

Problem:

A cylindrical shell rolls without slipping down an incline as shown below.  If it starts from rest, how far must it roll along the incline to obtain a speed v?

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Solution:

Problem:

A cylinder of mass m, radius r, and length l rolls straight down an inclined plane of length L and angle 30° with respect to the horizontal.  A sphere of mass M, and radius R rolls down another inclined plane, also of length L but at an angle θ with respect to the horizontal.  Each object is solid, of uniform density, and rolls without slipping.  Find an expression for the angle θ for which the two objects reach the bottom at the same time after they are released from rest at the same time.

Solution:

Problem:

An inclined plane of length L makes an angle θ > 0o with respect to the horizontal.  A very thin ring with radius R rolls down this plane.  It reaches the bottom in time Δtring.
(a)  If a uniform density disk or sphere with the same radius R roll down this plane, at what time do they reach the bottom?  Find Δtdisk/Δtring and Δtsphere/Δtring.
(b)  If all three object should reach the bottom at the same time Δtring, one needs to vary the angle θ for the disk and the sphere.  Find sinθdisk/sinθ and sinθsphere/sinθ.

Solution:

Problem:

Two identical uniform cylinders of radius R each are placed on top of each other next to a wall as shown.  After a disturbance, the bottom cylinder slightly moves to the right and the system comes into motion.  Find the maximum subsequent speed of the bottom cylinder.  Neglect friction between all surfaces.

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Solution: