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

__Forces - Problems__

*Problem 1*

A car of mass

**uniformly accelerates from a state of rest to a maximum speed**

*m***along a straight road during time**

*v*_{max}**,**

*t*_{max}(a) What is the most convenient reference frame to solve this problem?

(b) What is the vector of acceleration

**as a function of time?**

*a(t)*(c) What is the force

**its engine produces to push car forward as a function of time?**

*F(t)*(d) What is the vector of velocity

**as a function of time?**

*v(t)*(e) What is the distance

**it covers as a function of time?**

*d(t)**Answers*:

(a) Reference frame:

Origin at the beginning position of a car on a road, X-axis goes along

the road in the direction of car's movement, Y- and Z-axes are in any

position perpendicular to X-axis.

In this frame the Y- and Z-coordinates will always be zero for any moment of time

**.**

*t*(b) Acceleration: vector directed along the X-axis with magnitude

*x"(t) = a = |a(t)| = v*_{max }/t_{max}(c) Force: vector directed along the X-axis with magnitude

*|F(t)| = m·a = m·v*_{max }/t_{max}(d) Velocity: vector directed along the X-axis with magnitude

*x'(t) = |v(t)| = a·t = v*_{max }·t /t_{max}(e) Distance:

*x(t) = d(t) = a·t²/2 = v*_{max }·t²/(2·t_{max})*Checking*:

Acceleration must be constant:

*x"(t) = v*_{max }/t_{max}(does not depend on

**)**

*t*At

**distance from the beginning position must be zero:**

*t=0*

*x(0) = v*_{max }·0²/(2t_{max}) = 0At

**vector of velocity must have magnitude of zero:**

*t=0*

*x'(0) = v*_{max }·0/t_{max}= 0At

**vector of velocity must have magnitude**

*t=t*_{max}**:**

*v*_{max}

*x'(t*_{max}) = v_{max }·t_{max}/t_{max}= v_{max }*Problem 2*

Exactly the same as

*Problem 1*above, except in the beginning at time

**a car is not at rest, but moves with speed**

*t=0***.**

*v*_{min}*Answers*:

(a) Reference frame:

Origin at the beginning position of a car on a road, X-axis goes along

the road in the direction of car's movement, Y- and Z-axes are in any

position perpendicular to X-axis.

In this frame the Y- and Z-coordinates will always be zero for any moment of time

**.**

*t*(b) Acceleration: vector directed along the X-axis with magnitude

*x"(t) = a = |a(t)| = (v*_{max}−v_{min})/t_{max}(c) Force: vector directed along the X-axis with magnitude

*|F(t)| = m·a = m·(v*_{max}−v_{min})/t_{max}(d) Velocity: vector directed along the X-axis with magnitude

*x'(t) = |v(t)| = v*_{min}+ a·t = v_{min}+ (v_{max}−v_{min})·t /t_{max}(e) Distance:

*x(t) = d(t) = v*_{min}·t + a·t²/2 = v_{min}·t + (v_{max}−v_{min})·t²/(2·t_{max})*Checking*:

Acceleration must be constant:

*x"(t) = (v*_{max}−v_{min})/t_{max}(does not depend on

**)**

*t*At

**distance from the beginning position must be zero:**

*t=0*

*x(0) = v*_{min}·0 + (v_{max}−v_{min})·0²/(2·t_{max}) = 0At

**vector of velocity must have magnitude of**

*t=0***:**

*v*_{min}

*x'(0) = v*_{min}+ (v_{max}−v_{min})·0 /t_{max}= v_{min}At

**vector of velocity must have magnitude**

*t=t*_{max}**:**

*v*_{max}

*x'(t*_{max}) = v_{min}+ (v_{max}−v_{min})·t_{max }/t_{max}= v_{max}*Problem 3*

A planet of mass

**uniformly (that is, with constant angular speed) rotates around its star on a constant distance**

*m***from it, making one full circle (that is,**

*R***radians) in**

*2π***time.**

*t*_{o}(a) What is the most convenient reference frame to solve this problem?

(b) What is the angular speed

**of a planet as it rotates around its star?**

*ω*(c) What is the position vector of a planet as it goes along its

circular trajectory in XY-coordinates (Z-coordinate is always 0 for

properly chosen frame of reference)?

(d) What is the vector of velocity of a planet as it goes along its

circular trajectory, and how is this vector directed relative to

position vector?

(e) What is the magnitude of velocity vector (speed)

**?**

*v*(f) What is the vector of acceleration of a planet as it goes along its

circular trajectory, and how is this vector directed relative to

position vector?

(g) What is the magnitude of acceleration vector

**?**

*a*(h) Express the magnitude of acceleration vector

**in terms of radius of a circular trajectory**

*a***and the magnitude of the velocity vector**

*R***.**

*v*(i) What is the vector of force of gravity from a star to a planet in

terms of mass, radius of trajectory and speed (angular and linear)?

*Answers*:

(a) Reference frame:

Origin at the center of a star, XY-plane should be a plane where a

planet rotates, Z-axis is perpendicular to XY-plane, X-axis is from a

star to a point where planet is located at moment of time

*.*

**t=0**In this frame the Z-coordinate will always be zero for any moment of time

**.**

*t*(b) Angular speed:

**- constant**

*ω = 2π/t*_{o}(c) Position vector {

**}:**

*x(t), y(t)*

*x(t) = R·cos(ωt)*

*y(t) = R·sin(ωt)*(d) Velocity {

**}**

*x'(t), y'(t)*(first derivative of position):

*x'(t) = −R·ω·sin(ωt)*

*y'(t) = R·ω·cos(ωt)*This velocity vector is perpendicular to position vector since their

*scalar product*

**equals to zero.**

*x(t)·x'(t)+y(t)·y'(t)*(e) Magnitude of velocity vector (speed):

*v = √x'(t)²+y'(t)² = R·ω*(f) Acceleration {

**}**

*x"(t), y"(t)*(second derivative of position or

first derivative of velocity):

*x"(t) = −R·ω²·cos(ωt)*

*y"(t) = −R·ω²·sin(ωt)*This velocity vector is collinear to position vector, but directed

towards the origin of coordinates because both components have negative

multiplier

**in front of them.**

*−ω²*(g) Magnitude of acceleration vector:

*a = √x"(t)²+y"(t)² = R·ω²*(h)

*a = v²/R*(i)

*F = m·a = m·R·ω² = m·v²/R*
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