Apply problem-solving techniques to solve for quantities in more complex systems of forces.
Integrate concepts from kinematics to solve problems using Newton's laws of motion.
There are many interesting applications of Newton’s laws of motion, a few more of which are presented in this section. These serve also to illustrate some further subtleties of physics and to help build problem-solving skills.
Drag force on a barge
Suppose two tugboats push on a barge at different angles, as shown in
[link] . The first tugboat exerts a force of
in the
x -direction, and the second tugboat exerts a force of
in the
y -direction.
If the mass of the barge is
and its acceleration is observed to be
in the direction shown, what is the drag force of the water on the barge resisting the motion? (Note: drag force is a frictional force exerted by fluids, such as air or water. The drag force opposes the motion of the object.)
Strategy
The directions and magnitudes of acceleration and the applied forces are given in
[link](a) . We will define the total force of the tugboats on the barge as
so that:
Since the barge is flat bottomed, the drag of the water
will be in the direction opposite to
, as shown in the free-body diagram in
[link] (b). The system of interest here is the barge, since the forces on
it
are given as well as its acceleration. Our strategy is to find the magnitude and direction of the net applied force
, and then apply Newton’s second law to solve for the drag force
.
Solution
Since
and
are perpendicular, the magnitude and direction of
are easily found. First, the resultant magnitude is given by the Pythagorean theorem:
The angle is given by
which we know, because of Newton’s first law, is the same direction as the acceleration.
is in the opposite direction of
, since it acts to slow down the acceleration. Therefore, the net external force is in the same direction as
, but its magnitude is slightly less than
. The problem is now one-dimensional. From
[link](b) , we can see that
But Newton’s second law states that
Thus,
This can be solved for the magnitude of the drag force of the water
in terms of known quantities:
Substituting known values gives
The direction of
has already been determined to be in the direction opposite to
, or at an angle of
south of west.
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits