A loop of wire is perpendicular to a magnetic field such that the field is coming straight at you through the loop. The field be
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Answer:
The maximum speed at which the car can safety travel around the track is 18.6m/s.
Explanation:
Since the car is in circular motion, there has to be a centripetal force . In this case, the only force that applies for that is the static frictional force
between the tires and the track. Then, we can write that:
And since and
, we have:
Now, if we write the vertical equation of motion of the car (in which there are only the weight and the normal force), we obtain:
Substituting this expression for and solving for
, we get:
Finally, plugging in the given values for the coefficient of friction and the radius of the track, we have:
It means that in its maximum value, the speed of the car is equal to 18.6m/s.
Answer:
a)
b) See the proof below
c)
Explanation:
Part a
For this case we have the following differential equation:
With the initial condition
We can rewrite the differential equation like this:
And if we integrate both sides we got:
Where is a constant. If we apply exponential for both sides we got:
Using the initial condition we got:
So then our solution for the differential equation is given by:
For the half life we know that we need to find the value of t for where we have if we use this condition we have:
Applying natural log we have this:
And then the value of t would be:
And using the fact that we have this:
Part b
For this case we need to show that the solution on part a can be written as:
For this case we have the following model:
If we replace the value of k obtained from part a we got:
And we can rewrite this expression like this:
And we can cancel the exponential with the natural log and we have this:
Part c
For this case we want to find the value of t when we have remaining
So we can use the following equation:
Simplifying we got:
We can apply natural log on both sides and we got:
And if we solve for t we got:
We can rewrite this expression like this:
Using properties of natural logs we got: