Quiz #2

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This quiz is due at the beginning of class on the 7th of November (Friday). You are on your honor not to consult others or reference materials in the completion of this quiz. This includes textbooks and notes. Please note that there are two sides to the quiz.

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1.) (15 Points) Consider a ballistic pendulum. It is a device used to measure the speed of small objects travelling with large velocities. It consists of a wooden block of mass M suspended from vertical strings. Suppose a bullet of mass m is fired towards our pendulum with a velocity v₀. The bullet embeds itself in the block. The instant the bullet first touches the block we let t = t₀. When the bullet comes to rest in the block, t = t₁. Then the block has some velocity at t₁ given by v₁, when the strings are still approximately vertical. Then the block rises a height h in the $\hat{y}$ direction and comes to rest momentarily, tracing out a circular path. At this point, t = t₂.

a) For the time interval t₀ to t₁, is momentum conserved? If so, then give an expression for the total momentum in this time interval. If not, then state why not and suggest a system where the momentum would be conserved.

Solution:

Yes, momentum is conserved during the time interval.

p = m v₀ = (M + m) v₁

b) For the time interval t₀ to t₁, is energy conserved? If not, where was the energy lost? Give an expression for the amount of energy lost. If so, then give an expression for the total energy in this time interval.

Solution:

No, energy isn't conserved. Energy is lost to friction and other non-conservative forces during the collision. Some of these forces go in to deformation of the block, heat, sound, etc.

c) For the time interval t₁ to t₂, is momentum conserved? If so, then give an expression for the total momentum in this time interval. If not, state why not and suggest a system in which the total momentum would be conserved.

Solution:

No. External forces are present (gravity). If we included the earth in the system such that the force of gravity was now an internal force, then the momentum of the system is conserved.

d) For the time interval t₁ to t₂, is energy conserved? If not, where was the energy lost? Give an expression for the amount of energy lost. If so, then give an expression for the total energy in this time interval.

Solution:

Yes, energy is conserved.

$E = \frac{1}{2} (m + M) v_{1}^{2} = m g h$

e) Find an expression for the initial velocity of the bullet as a function of the height h and the masses involved.

Solution:

From part a) we know that

$v_{1} = \frac{m}{M+m} v_{0}$

Substituting this in to our equation that we got in d) we find that

$v_{0} = \frac{m+M}{m} \sqrt{2gh}$

2) (10 Points) A spring with force constant k hangs vertically. A block of mass m is attached to the unstretched spring and allowed to fall from rest. Find the distance the block falls before it begins moving upward again. Partial credit will be given for giving the proper values of the energy of the system at some useful initial and final times.

Solution:

The initial energy of the system is zero. The final energy of the system is given by

E_f = U_f + K_f

$E_{f} = \frac{1}{2} k y^{2} - mgy + \frac{1}{2} m v_{f}^{2}$

If we want the lowest value for y, we let $v_{f} \rightarrow 0$. We then find that

$E_{f} = \frac{1}{2} k y_{max}^{2} - mgy_{max}$

Setting this equal to the initial energy E₀ = 0, we find that

$\frac{1}{2} k y_{max}^{2} - mgy_{max} ~=~ 0$

Solving for y_max we get that

$y_{max} ~=~ \frac{2mg}{k}$