*Notes to a video lecture on http://www.unizor.com*

__Superposition of Forces -__

Problems 2

Problems 2

*Problem A*

Consider a slope that makes an angle

**with the ground. On**

*θ*the top of this slope there is a pulley, through which goes a

weightless non-stretchable thread that connects two point-objects:

one of mass

**, sliding up the slope, and**

*M*_{1}another of a bigger mass

**, hanging off the top pulley vertically down and pulling the thread down, thus forcing the smaller object to go up the slope.**

*M*_{2}Determine the direction and magnitude of the pressure vector

**on the pulley's axis.**

*P**Answer*:

Direction of the vector of pressure on the pulley's axis

**is along the angle bisector of the angle at the top of a pulley.**

*P*The magnitude of this vector is

*P = k·[1+sin(θ)]·cos(π/4−θ/2)*where

*k = 2M*_{1}·M_{2}·g / (M_{1}+M_{2})*Problem B*

A point-object is hanging on a thread of length

**and is uniformly moving along a circular trajectory in a horizontal plane, while a thread always maintains a constant angle**

*L***with a vertical.**

*φ*How many loops

**this object does in time**

*N(t)***?**

*t**Answer*:

*N(t) = (t/2π)·√g/[L·cos(φ)]*where

**is a free fall acceleration.**

*g**Problem C*

A wedge of mass

**and angle**

*m***slides down between two point-objects of mass**

*2φ***each, pushing them apart without any friction.**

*M*Find the acceleration

**of each of the two objects that a wedge pushes apart.**

*a**Answer*:

*a = g·tan(φ)/*

/[1+2(M/m)·tan²(φ)]/[1+2(M/m)·tan²(φ)]

*Problem D*

During long jump competition a sportsman accelerates with constant acceleration

**during time**

*a***and then jumps vertically up.**

*t*During his jump he goes up to height

**, while continuing moving forward because of his initial horizontal speed he gained before jumping.**

*h*What is the length

**of his jump?**

*L**Answer*:

*L = 2a·t·√2h/g*

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