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Problem 1: Quick Multiple Choice Questions [10 pts]
For each of the following questions circle the correct answer. Note that each question is worth only 2 points, so do not spend a lot of time on this part!
(a) Compared to the gravitational force with which the Earth pulls you, the gravitational force with which you pull the Earth is
Equal You exert no gravitational Greater Less force on the Earth
(b) Given a force law
[M][L]-1[T]-
, what are the dimensions of A?
[M][L][T]-2^ [L][T]-1^ [M][L]-2[T]-
(c) A metal hammer and rubber mallet of identical masses are swung at a nail with identical speeds. Which applies the greater impulse?
Hammer Mallet The impulses are the same
(d) A block with mass M and contact area A slides down an inclined plane with friction, covering a distance L in time T. How much time does it take another block with the same mass and composition, but contact area 2A, to slide down the same length?
(e) A pendulum of length L supporting mass M swings back and forth with period P. If the mass is doubled, what is the new period?
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Problem 2: Blocks and Pulley on an Incline [20 pts]
A block of mass M sits on an inclined plane, and is connected via a massless string through a massless pulley A (that slide without friction on the plane) to a fixed post. This pulley is in turn connected via a massless string through a second massless pulley B (attached to the top of the inclined plane and oriented to rotate about a horizontal axle) to a second block of mass 2M that hangs above the ground. The coefficient of static friction between the inclined plane and block resting on the inclined plane is 0 < < 1. Gravity is assumed to be acting in a
vertical direction with constant acceleration. The inclined plane is tilted to an angle with respect to horizontal, and the masses are assumed to be initially at rest.
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(b) [15 pts] Derive a relation, as a function of alone, for the minimum angle that the inclined plane can be tilted before the blocks start to move. You are
not asked to solve explicitly for in the final relation.
This problem can be treated as a statics problem; i.e., if none of the
blocks are moving, the net forces must equal 0. We first write down the
forces acting on the blocks using an inclined coordinate system for the
mass on the inclined plane (so is parallel to the surface), and a vertical
coordinate for the hanging mass ( ). Then:
Plugging back into the first equation we derive:
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Problem 3: Saving Yourself from a Fall [25 pts]
An intrepid student of mass walks onto a platform of mass that is attached to the side of a cliff of height H. When the student reaches the center of the platform the support breaks and both the student and platform plunge to the ground below. However, just before impact, the student jumps off of the platform with
sufficient force that she reaches zero velocity with respect to the ground ( ), thereby saving herself from injury. In this problem, assume that the acceleration due to gravity is a constant , and that the viscosity of air is negligible.
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(b) [10 pts] Assuming that the student can jump with a velocity with respect to the platform (at the end of her jump), derive an expression for her speed, , with respect to the (unmoving) ground. Assume that the jump is instantaneous.
The velocity of the student in the frame of rest is the jump velocity
relative the velocity of the platform after it has accelerated to its new
speed, ; i.e.,
If the jump is instantaneous, then the impulse on the system vanishes
and we have momentum conservation:
The initial and final momenta are:
and hence
Substituting this back in to the equation for vs above gives
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(c) [10 pts] To the limit that the platform is much more massive than the student
), what is the maximum height the student can fall and still jump to zero velocity with respect to the ground? Assume that the student can normally jump to a height of 1 foot from the ground.
Jumping from the ground, the equation of motion for a student is again
ballistic:
the initial conditions are and , so
At the maximum height, the vertical velocity vanishes so
In relation to the maximum height h the student can jump:
Plugging this into the above solution for and
implies:
1 foot
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[20 pts] Show that the student will execute simple harmonic motion as she falls through the tunnel. Find the time it takes the student to return to her point of departure and compare this to the time needed for a satellite to circle the Earth in a low orbit with r ≈ R. Ignore any effects due to the Earth’s rotation (N.B.: This is similar to problem 2.26 in K&K).
The gravitational force acting on the
person is Newton’s inverse square law:
where
This acts toward the center, but as we are
interested in the motion along the tube,
we need to use the variable
(see figure). The component of the gravitational force in the
direction of the tube is also dependent on so we get:
where (since G, M and R are all positive numbers). This is
the equation for simple harmonic motion.
The period of the motion (which equals the time it takes for someone
who has fallen in to come back to the same spot) is:
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For a satellite in low earth circular orbit, the gravitational acceleration
balances centripetal acceleration:
So we derive the same period:
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(a) [10 pts] Find the equations of motion for the rocket in polar coordinates, using the parameters specified above. Which component of momentum is conserved (assuming no net torque on the rocket)?
The rocket equation in vector notation is:
We need to use the form of acceleration in polar coordinates:
plugging this in we get:
Where I have used. The time dependent mass is
, and breaking the solution by components gives:
The component of momentum is conserved, as the external force (N)
acts only in the direction.
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(b) [10 pts] What is the speed of the rocket, , as a function of time?
Using the component of the equations of motion gives
Now we do a substitution to solve the integral
And arrive at:
Since , we get
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