I need help seeing if my thoughts are correct and how to do some things.
Block 1 of mass 0.200 kg is sliding to the right over a frictionless elevated surface at a speed of 8.00 m / s. The block undergoes an elastic collision with stationary block 2, which is attached to a spring of spring constant 1208.5 N / m. (Assume that the spring does not affect the collision.) After the collision, block 2 oscillates in SHM with a period of 0.140 s, and block 1 slides off the opposite end of the elevated surface landing a distance d from the base of that surface after falling height h = 4.90 m.
(a) Write an expression that gives the displacement of block 2 as a function of time. This expression must include the values of the amplitude of vibration and the angular frequency.
I came up with
x(t)= Acos(omega*t + pi/2)
not sure about +/- for the angle though and how you know which sign to have.(need help on this determination)
I found omega and m2 through: T= 2pi/omega= 2pi sqrt(m/k)
m1v1i + m2v2i = m1v1f + m2v2f
and found v2f and v1f
I was thinking that the v1f I found was the same velocity that the block 1 leaves with and travels off the table with IS THIS CORRECT?
Then I was thinking of using the v1f and v2f in energy equation to find the distance that the spring compresses (Amplitude) so I can plug it into the equation for cos
1/2mv1f + 1/2m2vf = 1/2kx^2
Is this alright?
b) Use differential calculus to obtain expressions for the velocity and acceleration of block 2 as functions of time.
once again I'm not sure if phi's angle or even if phi is correct.
x(t)= A cos (omega*t + pi/2)
v(t)= -omega A sin( omega*t + pi/2)
a(t)= -omega^2 cos (omega*t + pi/2)
c) What are the displacement, velocity, and acceleration of block 2 at t = 0.520 s?
I think I'd just plug into the equation after I find the values from a
(d) What is the value of d?
I know this is projectile motion problem with I think v in x direction...but if it is then would an angle be included? I think yes but I haven't worked with many problems with a object falling after sliding off a level surface.
how would I approach this?
Thank you drwls =)
a) The displacement of block 2 will be 0 at t=0, and it will vibrate about this position. They already tell you the period is P = 0.140 s. Figure out tne mass m2 from the relation
P = 2 pi sqrt (m2/k) = 0.140 s
m2/k = [P/(2 pi)]^2 = 4.965*10^-4.
I get m2 = 0.600 kg, 3 times the mass of m1.
You need to compute the amplitude of vibration to complete this part. You can get that by using energy and momentum conservation to compute the velocity of m2 right after collision. I believe you will find this to be 4.00 m/s. Vmax of mass2 equals omega*A, where A is the amplitude. In your case,
omega = 2 pi/P = 44.88 rad/s
Therefore A = (4.00 m/s)/44.88 rad/s) = 8.91*10^-2 m
Another way to get the amplitude is to set (1/2) k A^2 equal to the kinetic energy of m2 right after the collision, which you suggested. It should give the same answer. Try it and see.
The displacement equation for mass 2 is, if I'm right,
X = 8.91*10^-2 m * sin(2 pi t/P)
= 8.91*10^-2 sin (44.88 t)
2) This step is straightforward since you know know X(t) -- assuming I did the calculations correctly. This you need to verify.
3) Yes, just plug in the numbers. Since that is exactly 3 periods later, you should get the same values you had at t=0
4) In doing the elastic collision problem, you should find that mass 1 bpunces back with a velocity of 4.0 m/s. The time it takes to fall a veritcal height of 4.90 m is
t = sqrt (2H/g) = sqrt 1 = 1.00 s
The distance d will therefore be 4.0 m from the base
X = 8.91*10^-2 m * sin(2 pi t/P)
= 8.91*10^-2 sin (44.88 t)
I don't get this...why did you use sin? and how did you get 2pi t/P ??
I used these formulas below but are they incorrect?
x(t)= A cos (omega*t + pi/2)
v(t)= -omega A sin( omega*t + pi/2)
a(t)= -omega^2 cos (omega*t + pi/2)
P.S.- I was also wondering if there is a good site on the web that can explain the relation of the position of a spring to the sin/cos function since I have problems visualizing the two and which one I should use for which situation.
I used sin because the displacement at time =0, and then it becomes positive at first.
cos (wt + pi/2) is the same thing as -sin wt, anyway.
Your formula is OK if you define positive motion to be opposite to the direction of m1 before impact.
However, in the first part I think they want you to provide the actual value of A.
2 pi t/P is the same thing as w t, since 1/P is the frequency f and
2 pi f = w
Thanks drwls
I was wondering though if I used sin then wouldn't the
v(t)= omega A cos (omega* t)?
and
a(t)= -omega^2 A sin(omega* t)?
Let's go through each question and break down the steps to find the answers.
(a) To find the expression for the displacement of block 2 as a function of time, you correctly identified the formula: x(t) = Acos(ωt + φ), where A is the amplitude of vibration and ω is the angular frequency. The angle φ, also known as the phase angle, determines the initial position of the block. In SHM, the angle φ is determined by the initial conditions of the motion. If the block starts at its equilibrium position, then φ = 0. The plus/minus sign in the cosine function depends on the initial direction of motion. If the block starts at its maximum displacement to the right, then the phase angle should be 0 and the cosine function should have a plus sign. If it starts at its maximum displacement to the left, then the phase angle should be π and the cosine function should have a minus sign.
To determine the correct sign for the angle, you need to analyze the initial conditions of the system in the problem statement. Since it is mentioned that block 1 is sliding to the right and collides with block 2, we can assume that block 2 is displaced to the right initially. Therefore, the correct expression would be: x(t) = Acos(ωt).
(b) To obtain expressions for the velocity and acceleration of block 2 as functions of time, you correctly used differential calculus. The displacement function is x(t) = Acos(ωt). By differentiating this function once with respect to time, you get the velocity function: v(t) = -ωAsin(ωt). By differentiating the velocity function with respect to time, you get the acceleration function: a(t) = -ω^2cos(ωt).
(c) To find the displacement, velocity, and acceleration of block 2 at t = 0.520 s, you need to substitute the given time into the expressions for x(t), v(t), and a(t) that you derived in part (b). Plug in t = 0.520 s and the known values of A and ω to calculate the values.
(d) To find the value of d, which represents the horizontal distance traveled by block 1 after it falls off the elevated surface, you should treat this as a projectile motion problem. Block 1 has an initial horizontal velocity from sliding off the surface. The vertical motion is influenced by the gravitational force. Start by solving for the time it takes block 1 to hit the ground after sliding off the surface using the formula for vertical motion: h = (1/2)gt^2, where h is the height and g is the acceleration due to gravity. Once you find the time, you can calculate the horizontal distance traveled using the horizontal velocity: d = v*t.
Remember to pay attention to the signs in the projectile motion equations since the direction of motion is likely to be in the x-axis.