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

__Kinematics Problems 2__

*Problem 1*

A gear consists of two toothed wheels connected to each other as on this picture:

The size of the teeth is negligible relative to the size of the wheels.

A bigger wheel rotates around fixed point

**, has radius**

*O***and angular speed**

*R***.**

*ω*A smaller wheel rotates around fixed point

**and has radius**

*A***.**

*r*Wheels are locked by their teeth, so the rotation of a bigger wheel forces the rotation of a smaller one.

What is the angular speed of rotation of a smaller wheel?

*Answer*:

*R·ω/r**Problem 2*

Assume a frame of reference with Z-axis along the Earth's axis and

XY-plane containing an equator, but fixed relative to stars, so the

Earth rotates in this frame around Z-axis.

Also assume that the Earth is an ideal sphere of a radius

*R=6400 km***.**

*T=24 hours*What is a magnitude

**of a vector of velocity (**

*v**linear speed*) of a point on the surface of the Earth on the latitude of

**?**

*φ=60°**Answer*:

**.**

*v = 2πR·cos(φ)/T ≅*

≅ 838 km/hour ≅ 233 m/sec≅ 838 km/hour ≅ 233 m/sec

*Problem 3*

Assume a frame of reference is attached to the Earth with Z-axis along the Earth's axis and XY-plane containing an equator.

Also assume that the Earth is an ideal sphere of a radius

*R=6400 km***.**

*T=24 hours*A plane flies at altitude

*H=10,000 m*

*v=900 km/hour*A passenger who always looks through a window sees the Sun at exactly the same position all the time.

What is the latitude

**, above which the plane is flying?**

*φ**Answer*:

*cos(φ) = (v·T)/*[*2π·(R+H)*]*≅*

≅ 0.536310≅ 0.536310

**.**

*φ ≅ 1.00473725 rad ≅*

≅ 57.5672°≅ 57.5672°

*Problem 4*

Two hands on the clock coincide at noon.

How long will it take for them to coincide again?

*Answer*:

**.**

*12/11 hour**Problem 5*

A river flows along a straight line with constant flow with both banks parallel to each other. The width of a river is S.

A boat goes from one river bank to another, maintaining a course

perpendicular to a river, with a speed twice the speed of the river

flow.

How far down the river the flow will take the boat from a point directly opposite to a starting point of a boat?

*Answer*:

**.**

*S/2**Problem 6*

There are two cars on XY-plane with Cartesian coordinates.

Car #1 at moment

**is at position (**

*t=0***) on the X-axis, where**

*A,0*

*A > 0***) on the Y-axis, where**

*0,B***.**

*B > 0*Both cars go along their respective axes towards the origin of coordinates with speeds

**for car #1 and**

*u***for car #2.**

*v*At what moment in time

**the distance between the cars will be minimal?**

*t*_{min}*Answer*:

**.**

*t*_{min}= (A·u + B·v)/(u² + v²)
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