Classical Mechanics Answers

Question 2

SpaceX plans to transport humans to Mars, but experienced a few rocket failures while on initial tests. So suppose the rocket to Mars (without people on board - another test flight) liftoffs from earth, from rest, with an acceleration of 52.0 m/s². Unfortunately, after 8.15 s, the rocket experiences a failure and shuts off. It then continues to coast upward (with no appreciable air resistance). What is the maximum height the rocket reaches above the surface?

Your doctor has told you to watch your weight. Of course, your weight comprises two factors, so you assume that she wants you to watch both your mass and the local value g. Which explains why you are carrying a small mass m on an elastic band wherever you go. Most of the time, the elastic band is stretched by a distance of 2.1 cm. However, when you have finished lunch, you go into a very tall building and take the high speed lift (or elevator depending on where you live). Soon after the doors close, you notice that the band (which has linear elastic properties) is stretched by a total distance of 3.5 cm.

a) At what rate is the lift accelerating?

b) Your doctor will want to know the factor by which your weight has increased after that lunch. Suppose that you stand on scales in the lift while it is accelerating as described above. According to the scales, your weight has increased by a factor of _____ .

Q8. Calculate the divergence and Curl, of the following functions

a. V= x2 i+3xy^2 j -2xyz k

b. V=xyi+2yz j+3zx k

c. V= y2i +(2xy+z2)j+ 2yz k , where i , j , k are unit vectors along x, y and z axis

Q7. The height of certain hill (in feet) is given by h(x,y)=10(2xy−3x2 −4y2 −18x+28y+12)

Where y is the distance in north, x is the distance in east.

a. Where is the top of the hill located?

b. How high is the hill?

c. How steep is the slope (in feet per mile) at a point 1-mile north and 1- mile east, in what

direction is the slope steepest?

Q6. Find the gradient of the following functions

f(x, y, z) = x2y3z4.

f(x, y, z) = exsin(y) ln(z).

A car moves at a constant acceleration and crosses a path of 400m. What is the acceleration if the car takes 10 seconds to cross the third 100m path?

A particle exhibits simple harmonic motion along the horizontal direction with angular frequency ω. Consider a point P, at a distance x from the particles equilibrium position. When the particle passes P, it has a velocity v. Show that the time in which it returns to the point P is given by tr = 1/w arctan [( 2vxω )/(ω2x 2 − v 2)]

The figure shows two equal masses connected by a spring with two other identical springs fixed to rigid supports on either side. This system permits the masses to jointly undergo simple harmonic motion along a straight line such that the system corresponds to two couple oscillators. (a) Explain clearly why the equations of motion of the two masses should be mx¨1 + k(2x1 − x2) = 0, mx¨2 + k(2x2 − x1) = 0 for mass m1 and mass m2 respectively. (b) Show that the system possesses two characteristic frequencies, one out of phase (fast) and the other in phase (slow), ωf = r 3k m and ωs = r k m . Note that there are numerous different ways of finding these frequencies from the equations of motion. These include assuming oscillatory motion of each mass, algebraically manipulating the equations of motion or solving the eigenvalue problem

A block of mass m is attached to a spring, with spring constant k, and a dashpot. The dashpot provides a retarding force with damping rate b and proportional to the velocity ˙x. When the block is a distance d0 from the left wall the spring is relaxed. When extended beyond d0 and released the block exhibits damped oscillatory motion. You may assume that gravitational effects are negligible. (a) Show that the equation of motion of this sytem has the form x¨ + 2γx˙ + ω 2 0x = 0 and identify the variables, γ and ω0. (b) Consider the case when γ < ω (under-damping). The block is now pulled to the right a distance x = a, from its equilibrium position, and at time t = 0 it is released from rest. Show that the subsequent motion is given by, x(t) = ae−γt  cos ω1t + γ ω1 sin ω1t  where ω1 = p ω 2 0 − γ 2. (c) At what time does the block first change its direction of motion after being released?