Solved Problems

3. Curvilinear Motion

Problem 2

A car driver enters a curve at 72 km/h, and slows down, making the speed decrease at a constant rate of 4.5 km/h each second. Make an estimate for the value of the radius of the curve using the scale shown in the figure. Find the acceleration of the car 4 seconds after the driver started to slow down.

Car slowing down in a curve

The radius is approximately 16.7 m. The tangential acceleration (decrease rate of the speed) is equal to

Solving the differential equation that relates the tangential acceleration to derivative of the velocity with respect to time, gives the velocity after 4 seconds (72 km/h equals 20 m/s)

And the total acceleration is

This value is just an approximation, because the radius has been obtained approximately by looking at the image.

Problem 6

Two cars A and B go through the curve shown in the figure following different paths. From a point on the line C, car B follows a semi-circumference of radius 102 m; until another point on line C. Car A moves from the line C following a straight line segment, it then follows a semi-circumference of radius 82 m and moves to another point on line C following another straight line segment. Both cars move at the highest speed that they can have without the tires sliding out of the circular path, which for the type of tires used means that the normal acceleration will have the maximum value of 0.8  , where is the acceleration of gravity. Find the times that both cars need to complete the curve, from the initial point on the line C, to the final point on the same line.

Two car in a curve

Since each car makes the curve with constant speed, the time it takes is for any of them is then

The speed of each car is that which leads to the maximum value of the normal acceleration, namely

In the case of car B, the length of the arc it follows is half the circumference of 102 meters radius

Hence, the time it takes is

In the case of car A, the length of the cruved path it follows beetween the two point in line C is half the circumference of 82 meters radius plus two straight segments of 20 m each

So the time it takes for car A is


Problem 7

A particle follows the path shown in the figure, starting from rest at A and then speeding up with constant acceleration until B. From B to E the speed of the particle remains constant at 10 m/s and at E the particle slows down, with constant acceleration, untill it stops at point F. The distance AB is 60 cm, CD is 20 cm and EF is 45 cm; arc BC has radius of 60 cm and arc DE has radius of 45 cm. Find:

  1. The magnitude of the acceleration of the particle in each of the paths AB, BC, CD, DE and EF.
  2. The total time of the motion from A to F and the average velocity in that motion.
Trajectory with two cruves

(a) Along the AB segment,

The magnitude of the acceleration is then 83.33 m/s2. On the EF segment,

The magnitude of the acceleration is 111.11 m/s2. In the segment CD, the acceleration is zero, because the motion is straight and uniform. In the arc BC, the acceleration has only normal component because the speed is constant:

Therefore, the magnitude of the acceleration is 166.67 m/s2. In the arc DE, the acceleration also has only normal component:

The magnitude of the acceleration is 222.22 m/s2.

(b) The total distance traveled is the sum of the three segments AB, CD and EF, plus the two arcs BC and DE, both with angle of radians:

The time the particle takes to travel the BCDE path is then:

To compute the time it takes to go through the AB segment, we integrate the differential equation for the tangential acceleration:

And the same procedure is used to compute the time it takes in the EF segment:

The average speed is equal to the distance traveled divided by the time it took:


Problem 8

A disk of radius 3 cm is glued to another disk of radius 6 cm, with a common axis, as shown in the figure. The horizontal bar A moves to the right at 10 m/s, keeping in contact with the bigger disk and without sliding on its surface. At the same time, the horizontal bar B moves to the left at 35 m/s, keeping in contact with the smaller disk and without sliding on its surface. Determine the direction of motion of the center O and find the velocity of O and the angular velocity of the disks.

Double wheel

Taking as positive the direction from left to right, the velocities of bars A and B are

And the velocity of the point of the wheel which is in contact with bar A, relative to the point of the wheel in contact with bar B is equal to

The positive sign of that result means that the wheel is rotating clockwise; its angular velocity is

In radians per second. The velocity of point O, relative to the point of the wheel in contact with bar B, is positive because the angular velocity is clockwise and its value is

Finally, the velocity of point O is

in units of m/s. The negative sign means that point O moves to the left.

Problem 9

A wheel of radius equal to 20 cm rolls without sliding over a plane horizontal surface, along the axis. At time the wheel's center is in and  cm and points P and Q in the wheel are in with and  cm. The velocity of the wheel's center remains constant and equal to 2 m/s. (a) Determine the time it takes the wheel to complete two turns. (b) Plot the trajectories of points P and Q in the interval of time corresponding to the two turns.

Points in a rolling wheel

(a) Since the velocity of point P is zero, the velocity of C relative to P is equal to 2 m/s and the angular velocity of the wheel is

Since it is constant, the time that the wheel takes to rotate two turns is

(b) The angle that the line CP does with the vertical is given by the expression

And the positions of points P and Q, relative to C, are then

The position of point C, as a function of time, is

Thus, the positions of points P and Q, as functions of time, are

The plot of the trajectories of these two points, during the two turns, can be done with the following Maxima command

(%i1) plot2d([[parametric, 2*t-0.2*sin(10*t), 0.2-0.2*cos(10*t)],
    [parametric, 2*t-0.1*sin(10*t), 0.2-0.1*cos(10*t)]], ;[t,0,1.26], [legend,"P","Q"]);

Cycloids

Problem 10

A dumbbell has two disks with radius of 6 cm, connected by a cylindrical bar with radius of 4 cm. A long strip of paper is taped and rolled around the bar as shown in the figure. The dumbbell is then placed on a flat horizontal surface and the paper strip is pulled horizontally with a constant speed of 2.5 cm/s, making the dumbbell roll over the surface without sliding.

  1. Find the angular velocity of the dumbbell.
  2. Explain in which direction will move the point O in the axis of the bar and the disks and find its velocity.
  3. How many centimeters of the paper strip will be wound or unwound per second?
Dumbbell rolling on a surface

Dumbbell

(a) Since the dumbbell rolls without sliding, the speed of point B is zero and the speed of A relative to B is then

in cm/s. That relative speed, divided by the distance between the projections of A and B in the rotation plane, gives the angular velocity of the dumbbell:

And the dumbbell rotates in the clockwise sense, since point A moves to the rigth, relative to point B.

(b) Since the dumbbell rotates in the clockwise sense, point O moves to the right and its speed is:

(c) The velocity of point C, relative to point O, is:

Being in the negative sense of the axis, it means that points O and C get closer so the strip winds even more around the dumbbell cylinder. Each second 5 cm of strip are wrapped around the cylinder.