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Problem 1

The truck shown in Figure P-1 moves from rest with constant acceleration ˙a up an incline of angle q. Find the greatest speed v that the truck can acquire in a distance d, if the crate C is not to slip on the truck bed. The coefficient of static friction is m.

Problem 2

In  Figure  P-2  a  horizontal  force  Ë™Fa  of  magnitude

20N is applied on the block as it slides a  distance of d = 0.5m up a frictionless ramp. Then,

 

(a)      During the displacement, find the net work done by Ë™Fa, the gravitational force and the normal force Ë™N on the block.

(b)      If the block has zero kinetic energy at the start, find the speed at the end of the displacement

Problem 3

Masses m2 and m3 are initially at rest and the spring of length â„“0 is unstressed. Then mass m1, traveling with velocity v0 in a direction perpendicular to the spring, hits m2 inelastically and sticks to it. In the ensuing motion the spring stretches to a maximum length 3â„“0 Solve for v0, assuming that the masses are equal and can be considered as particles.

 

 

 

Problem 4

Consider the conservative for

Figure P-2

Figure P-3


 

Ë™F = (2 z2 cosh x y2)Ë™i + 2 y (z x)Ë™j + (4 z sinh x + y2)Ë™k

(a)      Find the corresponding potential energy function.

(b)      Find the work done by Ë™F in moving a particle from the origin along the path y = sin(p x/2)

to the point (2, 0, 0).


Problem 5

A collar of mass m slides without friction along a circular track of radius R as shown in Figure P-5. Attached to the collar is a linear spring with spring constant K and unstretched length zero. The spring is attached at the fixed point A located a distance 2R from the center of the circle.  Assuming no gravity

and the initial conditions q(t = 0) = qand qË™ (t  =

0) = q˙0, determine

(a)      the differential equation of motion for the col- lar in terms the angle q

(b)      the reaction force exerted by the track on the collar as a function of the angle q

Problem 6

 

A star camera on a satellite observes three stars. The directions of the stars in the camera S frame are given by the unit vectors

S˙s1=

0.036103

0.20475


 

SË™s2 =


0.088521

3

 
0.24321

 


 

SË™s3 =


0.32139

9

 
0.11698

 


5

 
  0.9781

 


 0.9659  ï£»


   0.9396 


The astronomical star catalog lists the coordinates of the same stars in the inertial frame N as

 


        0.5591


   0.27356


   0.27022


5

 

2

 
N Ë™s1 =

 


0.34066


N Ë™s2 =


0.05727


N Ë™s3 =


0.57125


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Engineering Mathematics

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