Rolling Marble on table

In summary: I did it and got the same answer as above. In summary, the problem is to calculate the time it took for a small steel marble to travel from L to P on a horizontal table, given that it leaves the table at M and falls to the ground at P. Using the equation p(t) = 0.5 * a * t^2 + v0 * t + p0, with a vertical acceleration of -9.8 m/s^2, the time was found to be 2.85 seconds. However, the online submission said the answer was incorrect and another method was suggested, finding the time it took for the marble to fall 40cm and using that to calculate the horizontal velocity. Another method involved
  • #1

Homework Statement



A very small steel marble is shown rolling at a constant speed on a horizontal table. The marble leaves the table at M, falls, and hits the ground at P. This is illustrated in the diagram below which is drawn to scale. Calculate the time it took the marble to travel from L to P.

plot-1.png


Homework Equations



Professor Posted This
rollanddrop.gif

Still don't understand

p(t) = 0.5 * a * t^2 + v0 * t + p0
where
p(t) is position at time t
a is acceleration
v0 is initial velocity at time 0
p0 is position at time 0

The Attempt at a Solution



Vertical acceleration due to gravity = -9.8 m/s^2 = -980 cm/s^2

px(t) = 100 = 0.5 * 0 * t^2 + v0 * t + 60 = v0 * t + 60
vertical
py(t) = 0 = 0.5 * -980 * t^2 + 0 * t + 40 = -490 * t^2 + 40

t=2.85s

v0 = 210.011

got an answer and online submition said it was wrong

Please help
 
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  • #2
Basically you need to find how long it took for the ball to fall 70cm, then you will know the horizontal velocity.

x = x0 + v0t + (1/2)at2
40 = 0 + 0 + (1/2)gt2
t = sqrt(80/980) ~ 0.286 sec // you wrote t=2.85s, missed a zero ;)

then v horizontal is:

x = x0 + v0t + (1/2)at2

100 = 40 + vht + 0

vh = 60/sqrt(80/980) = 210 cm/sec


** The question your Proff. explained has the same idea but different numbers.
 
  • #3
There are perhaps two ways to do a question like this. One is to use physical insight to see the essence of the problem so you don't have to think too much about what signs values should be and so on. Another way is to leave nothing to chance and do everything the way it is given.

If you do the first, you would consider that you need the total time, which is just the sum of the two times from L to M and M to P. Each can be done independently, just to be added together afterwards.

Time L to M can't be done just yet because you don't know the speed of the marble. So start with time M to P. That is just the time it took the marble to fall 40cm.

Once you have that, you need to find the speed of the marble, which you can find using the time from M to P.

So you see, this is decomposing the problem into managable chunks, then doing each easy piece.


The other way is as follows but uses less physical insight. I think it is less nice:

At time t = 0, the marble is at L.
At time t = t_m, the marble is at M.
At time t = t_p, the marble is at P.

At time t = t_m:

s_x = 20 + v_x t_m = 40
s_y = 40

At time t = t_p:

s_x = 40 + v_x (t_p - t_m) = 100
s_y = 40 - (1/2) g (t_p - t_m)^2 = 0

We have three equations in three unknowns, v_x, t_m, t_p:

20 + v_x t_m = 40
40 + v_x (t_p - t_m) = 100
40 - (1/2) g (t_p - t_m)^2 = 0

Solve simultaneously to find t_p.
 

1. How does the marble roll on the table?

The marble rolls on the table due to the force of gravity acting on it. When the marble is placed on the table, it is pulled towards the center of the Earth by the force of gravity. As it moves, the friction between the marble and the table causes it to roll.

2. What factors affect the speed of the rolling marble?

The speed of the rolling marble is affected by several factors, including the angle at which it is released, the surface of the table, and the shape and weight of the marble. The steeper the angle of release, the faster the marble will roll. A smooth surface with low friction will also increase the speed of the marble, while a rough surface will slow it down. The weight and shape of the marble also play a role, with heavier and more spherical marbles rolling faster.

3. Why does the marble eventually stop rolling?

The marble eventually stops rolling due to the force of friction. As it rolls, the friction between the marble and the table slows it down, eventually bringing it to a stop. Other factors such as air resistance and imperfections on the surface of the table can also contribute to the marble's eventual stop.

4. What would happen if the marble rolled on a tilted surface?

If the marble rolled on a tilted surface, it would accelerate in the direction of the tilt due to the force of gravity. The steeper the angle of the tilt, the faster the marble would accelerate. The marble would continue rolling until it reached the bottom of the tilted surface or encountered an obstacle that stopped its motion.

5. Can you predict the path of the rolling marble?

Yes, the path of the rolling marble can be predicted using the laws of physics and mathematics. By considering the initial conditions, such as the angle of release and the weight of the marble, and taking into account factors like gravity and friction, it is possible to calculate the trajectory of the marble and predict its path.

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