Newton's Laws Applications Review
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Part D: ProblemSolving
22. A 945kg car traveling rightward at 22.6 m/s slams on the brakes and skids to a stop (with locked wheels). If the coefficient of friction between tires and road is 0.972, determine the distance required to stop. PSYW
Answer: 26.8 m
Like most problems, this problem begins with a freebody diagram (as shown at right). Note that there is no rightwards applied force (a common mistake). Note also that the force of friction is the only force responsible for the acceleration (deceleration) of the car. The F_{frict} value is the net force. So determining the acceleration involves finding F_{grav} (F_{grav}=m•g = 945•9.8 = 9261 N), F_{norm} (the same as F_{grav}= 9261 N), and F_{frict} (F_{frict} = mu•F_{norm} = 0.972 • 9261 N = 9002 N). With the ∑F_{x} = 9002 N, the acceleration can be calculated (a_{x} = ∑F_{x}/m = 9002 N/945 kg = 9.53 m/s/s.) This is a leftwards acceleration; a_{x} will be assigned the numerical value of 9.53 m/s/s in the next part of this problem.
Now in the kinematic part of this problem, the distance must be found using the known information (v_{f }= 0 m/s, v_{o} = 22.6 m/s and a = 9.53 m/s/s). Use the equation:
v_{f}^{2} = v_{o}^{2} + 2 a d
(0 m/s)^{2 }= (22.6 m/s)^{2} + 2•(9.53 m/s/s)•d
d = [(22.6 m/s)^{2}] / [2•(9.53 m/s/s)] = 26.8 m

[ #22  #23  #24  #25  #26  #27  #28  #29  #30 ]
23. A student pulls a 2kg backpack across the ice (assume frictionless) by pulling at a 30.0 degrees angle to the horizontal. The velocitytime graph for the motion is shown. Perform a careful analysis of the situation and determine the applied force. PSYW
Answer: 0.289 N
The acceleration of the object can be found from the slope of the line on the vt graph. The acceleration is:
a = change in velocity/change in time = (2.0 m/s) / 16 s = 0.125 m/s/s.
The net force is found from m•a; so F_{net} = (2 kg)•(0.125 m/s/s) = 0.250 N.
The only force that can contribute to the net force is the applied force. The horizontal component of the applied force must therefore be equal to 0.250 N. A diagram depicting the relationship of the lengths of the side for this 306090 triangle is shown at the right. The cosine function can be utilized to determine the magnitude of the tension force.
cos(30 deg) = 0.250 N)/F_{app}
Solving for F_{app} yields 0.289 N.

24. After its most recent delivery, the infamous stork announces the good news. If the sign has a mass of 10.0 kg, then what is the tensional force in each cable?
Answer: 56.6 N
If the mass of the sign is 10.0 kg, then the weight of the sign is 98 N (F_{grav} = m•g). This downward force of gravity must be balanced by the upward component of the tension. Thus, each cable must pull upwards with 49.0 N of force. In addition to the upward pull by the cable, there is also a horizontal pull. The tension force is the combined effect of these two componens. To determine the tension, set up a 603090 triangle with a vertical component of 4.09 N as shown at the right. Then, use the sine function to solve for the hypotenuse of the triangle. The work is shown below:
sin(60 deg.) = 49.0 N/F_{tens}
F_{tens} = (49.0 N)/(sin 60 deg) = 56.6 N

25. Splash Mountain at Disney World in Orlando, Florida is one of the steepest water plume rides in the United States. Occupants of the boat fall from a height of 100 feet (3.2 ft = 1 m) down a wet ramp which makes a 45 degree angle with the horizontal. Consider the mass of the boat and its occupants to be 1500 kg. The coefficient of friction between the boat and the ramp is 0.10. Determine the frictional force, the acceleration, the distance traveled along the incline, and the final velocity of the boat at the bottom of the incline. PSYW
Answer: All numerical answers are rounded to two significant digits: F_{frict} = ~1040 N; a = ~6.2 m/s/s; d = ~44 m; v_{f} = 23 m/s
This is a multipart inclined plane problem. Like all problems, it should begin with a freebody diagram. If this is where the difficulty lies for you, then take some time to review inclined planes. See help page. The friction force opposes the motion of the boat. If the + x direction is defined as down and along the incline (as shown at the right) then the net force can be computed by the expression F_{}  F_{frict}. The parallel component of the weight vector (m•g•sin theta) is 10395 N. The F_{frict }force is computed by multiplying the coefficient of friction (0.10) by the normal force (F_{norm}). The normal force is balancing F_{perp} (m•g•cos theta) and is equal to it. So F_{norm} = 10395 N and F_{frict} = 1039.4 N. The net force (F_{net} = F_{}  F_{frict }) is 9356 N; and the acceleration is F_{net}/m or 6.24 m/s/s.
Once the Newton's laws analysis has been completed and the acceleration determined, the kinematics portion of the problem can be tackled. First, determine the distance along the incline from the height of the hill and the angle of incline. The height of 100 ft is equivalent to 31.25 m; the distance along the incline is 44.19 m [found from (31.25 m)/sin (45)]. The final velocity is found using the acceleration, distance and a kinematic equation:
v_{f}^{2} = v_{o}^{2} + 2•a•d
v_{f}^{2} = (0 m/s)^{2} + 2•(6.24 m/s/s)•(31.25 m) = 551.28 m^{2}/s^{2}
v_{f} = 23 m/s

26. At last year's Homecoming Pep Rally, Trudy U. Skool (attempting to generate a little excitement) slid down a 42.0 degree incline from the sports dome to the courtyard below. The coefficient of friction between Trudy's jeans and the incline was 0.650. Determine Trudy's acceleration along the incline. Begin with a freebody diagram. PSYW
Answer: 1.82 m/s/s
This is nearly an identical force analysis as the last problem except the mass is not known. When the mass is not known, the problemsolving strategy involves inserting m into the equation as an unknown variable and proceding with the solution. It is likely that in a subsequent step of the problem that the m will cancel and the accceleration can be determined without knowing m.
The net force is found by the expression F_{}  F_{frict}. The parallel component of the weight vector is m•g•sin(theta). The F_{frict }force is computed by multiplying the coefficient of friction (mu) by the normal force (F_{norm}). The normal force is balancing F_{perp} and is equal to m•g•cos(theta). So F_{norm} = m•g•cos(theta) and F_{frict} = µ•m•g•cos(theta). The net force is m•g•sin(theta)  µ•m•g•cos(theta). The acceleration is F_{net}/m or [m•g•sin(theta)  µ•m•g•cos(theta)]/m. Note that there is an m in both terms in the numerator and a m in the denominator. The masses cancel and the equation reduces to
a = [g•sin(theta)  µ•g•cos(theta)]
By substitution of the given values into the equation, it can be shown that a = 1.82 m/s/s.

[ #22  #23  #24  #25  #26  #27  #28  #29  #30 ]
27. Baldwin Young is conducting his famous toupee experiments. He tips his head at a given angle and determines the coefficient of static and kinetic friction between a toupee (which is probably his own) and his scalp. At an angle of just barely 17.5 degrees, the toupee begins to accelerate from rest. Then Baldwin lowers the angle to 13.8 degrees to observe that the toupee moves with a constant speed. Use this information to determine the coefficients of both static and kinetic friction. Begin with a freebody diagram. PSYW
Answer: µ_{static} = 0.315; µ_{kinetic} = 0.246
A freebody diagram and a force analysis will yield the equation
F_{frict} = F_{}
for both the static and the kinetic situation. That is, when the object is at rest or moving down the incline at constant speed, the force down and parallel to the incline is balanced by the force which is up and parallel to the incline. The expressions for F_{frict} and F_{} can be substituted into this equation to yield
µ•F_{norm} = m•g•sin(theta)
Knowing that F_{norm} is equal in magnitude to F_{perp}, the expression for F_{perp} can be substituted into the equation for F_{norm}. The new equation becomes
µ•m•g•cos(theta) = m•g•sin(theta)
The angles for both the static and the kinetic case are known and mu (coefficient of friction) is the unknown. So the equation can be rearranged to
µ = m•g•sin(theta)/m•g•cos(theta)
The mass (m) and the acceleration of gravity (g) cancel from both the numerator and the denominator leaving sin(theta)/cos(theta) on the right side. This can be simplified again to
µ = tan(theta)
Equation 1 can be used to find the coefficient of friction for both the static and kinetic situation. The answers are as follows:
µ_{static} = tan(17.5 deg.) = 0.315
µ_{kinetic} = tan(13.8 deg.) = 0.246

[ #22  #23  #24  #25  #26  #27  #28  #29  #30 ]
28. Consider the twobody system at the right. There is a 0.250kg object accelerating across a rough surface. The sliding object is attached by a string to a 0.100 kg object which is suspended over a pulley. The coefficient of kinetic friction is 0.183. Calculate the acceleration of the block and the tension in the string. PSYW
Answer: a = 1.52 m/s/s ; F_{tens }= 0.83 N
The solution to this problem begins by drawing a freebody diagram for each object. Note that the + xaxis for the 0.25kg object is drawn in the direction that the object accelerates; and the + yaxis for the 0.10kg object is drawn in the direction which it accelerates.
Note that while friction acts upon the 0.25kg object, it does not act upon the 0.10kg object (since it is not being dragged across the surface). Newton's second law of motion (·F = m•a) can be applied to the motion of the 0.25kg mass:
∑F_{x} = m•a_{x}
F_{tens}  F_{frict }= m•a_{x}
The expressioin for F_{frict} can be substituted into the equation.
F_{tens}  µ•F_{norm }= m•a_{x}
Since the vertical forces on the 0.250kg object balance each other, it is known that F_{grav} = F_{norm}; so F_{norm} = m•g = (0.250 kg)•(9.8 m/s/s) = 2.45 N. Since µ = 0.183, µ•F_{norm} = 0.448 N. Equation 1 can now be rewritten as
F_{tens}  0.448 N_{ }= (0.250 kg)•a_{x}
Newton's second law of motion (∑F = m•a) can be applied to the motion of the 0.10okg mass:
∑F_{y} = m•a_{y}
F_{grav}  F_{tens} = m•a_{y}
The expression for F_{grav} (m•g) can be substituted into the equation.
m•g  F_{tens} = m•a_{y}
(0.100 kg)•(9.8 m/s/s)  F_{tens} = m•a_{y}
0.98 N  F_{tens} = m•a_{y}
The mass of the 0.100kg object can be substituted into the equation to yield equation 3.
0.98 N  F_{tens} = (0.100 kg)•a_{y}
Equations 2 and 3 each include an acceleration term  a_{x} and a_{y}. These acceleration values are the same since the system accelerates together; thus, a_{x }= a_{y} = a. An inspection of these two equations show that there are now two equations and two unknowns  F_{tens} and a. These unknown values can be solved for in the customary manner.
First, equation 3 is rearranged to create an expression for F_{tens}:
0.98 N  (0.100 kg)•a_{ }= F_{tens}
This expression for F_{tens} is now substituted into equation 2.
F_{tens}  0.448 N_{ }= (0.250 kg)•a
0.980 N  (0.100 kg)•a  0.448 N_{ }= (0.250 kg)•a
This equation can now be solved for acceleration (a) as shown in the following steps.
0.980 N  0.448 N_{ }= (0.250 kg)•a + (0.100 kg)•a
0.532 N = (0.305 kg)•a
a = 1.52 m/s/s
Now that a is known, its value can be substituted into equation 4 in order to solve for F_{tens}.
0.980 N  (0.100 kg)•a_{ }= F_{tens}
0.980 N  (0.10 kg)•(1.52 m/s/s)_{ }= F_{tens}
0.980 N  0. 152 N = F_{tens}
0.830 N = F_{tens}

29. Consider the twobody system at the right. A 22.7N block is placed upon an inclined plane which is inclined at a 17.2 degree angle. The block is attached by a string to a 34.5N block which is suspended over a pulley. The coefficient of friction is 0.219. Determine the acceleration of the block and the tension in the string. PSYW
Answer: a = 3.95 m/s/s; F_{tens} = 20.6 N
Like #21 and #28, this problem can most easily be solved using separate freebody analyses on the individual masses. Freebody diagrams, the chosen axes systems, and associated information is shown below. Note that in chosing the axis system, it has been assumed that object 1 will accelerate up the hill and object 2 will accelerate downwards on the back side of the hill. If this ends up to be false, then the acceleration values will turn out to be negative values.
The 22.7N object is on an inclined plane. The usual circumstances apply; there is no acceleration along what has been designated as the yaxis.
F_{norm} = F_{perp} = m•g•cos(theta) = 21.7 N
The parallel component of F_{grav} is
F_{} = m•g•sin(theta) = (22.7 N)•sin(17.2)
F_{} = 6.71 N
The F_{frict} value can be found from the F_{norm} value:
F_{frict} = µ•F_{norm} = (0.219)•(21.7 N) = 4.75 N
The ∑F_{x} = m•a_{x} equation can now be written:
∑F_{x} = m•a_{x}
F_{tens}  F_{frict } F_{} = m•a_{x}
F_{tens}  4.75 N_{ } 6.71 N = (2.32 kg)•a_{x}
(Note that the F_{frict} and F_{} forces are subtracted from F_{tens} since they are heading in the direction of the negative xaxis.)
The above process can be repeated for the 34.5N object. The ∑F_{y} = m•a_{y} equation can now be written:
∑F_{y} = m•a_{y}
F_{grav}  F_{tens } = m•a_{y}
(34.5 N)  F_{tens } = (3.52 kg)•a_{y}
The separate freebody analyses have provided two equations with two unknowns; the task at hand is to use these two equations to solve for F_{tens} and a.
Equation 2 can be rewritten as
(34.5 N) _{ }(3.52 kg)•a_{y} = F_{tens}
Since both objects accelerate together at the same rate, the a_{x} for the 22.7N object is equal to the a_{y} value for the 34.5N object. The subscripts x and y can be dropped and a can be inserted into each equation.
(34.5 N)  (3.52 kg) •a = F_{tens}
Equation 3 provides an expression for F_{tens} in terms of a. This expression is inserted into equation 1 in order to solve for acceleration. The steps are shown below.
F_{tens}  4.75 N_{ } 6.71 N = (2.32 kg)•a
(34.5 N)  (3.52 kg) •a  4.75 N_{ } 6.71 N = (2.32 kg)•a
23.0 N = (5.84 kg)•a
3.95 m/s^{2} = a
The value of a can be reinserted into equation 3 in order to solve for F_{tens}:
F_{tens} = (34.5 N)  (3.52 kg) • (3.95 m/s^{2})
F_{tens} = 20.6 N

30. A 10.0gram mass is tied to a string. The string is attached to a 500.0gram mass and stretched over a pulley, leaving the 10.0gram mass suspended above the floor. Determine the time it will take the 10.0gram mass to fall a distance of 1.50 meters if starting from rest. (No friction)
Answer: t = 3.95 s
This problem involves a blending of Newton's laws and kinematics. Newton's laws will have to be used to determine the acceleration; then kinematics will have to be used to determine the time to fall 1.50 m from rest. The solution to this problem begins by drawing a freebody diagram for each object. Note that the + xaxis for the 500.0gram object is drawn in the direction that the object accelerates; and the + yaxis for the 10.0gram object is drawn in the direction which it accelerates.
Note that while friction acts upon the 500.0gram object, it does not act upon the 10.0gram object (since it is not being dragged across the surface). Newton's second law of motion (∑F = m•a) can be applied to the motion of the 0.5000kg mass:
∑F_{x} = m•a_{x}
F_{tens}_{ }= m•a_{x}
F_{tens} = (0.5000 kg)•a_{x}
Newton's second law of motion (∑F = m•a) can also be applied to the motion of the 0.0100kg hanging mass:
∑F_{y} = m•a_{y}
F_{grav}  F_{tens} = m•a_{y}
The expression for F_{grav} (m•g) can be substituted into the equation.
m•g  F_{tens} = m•a_{y}
(0.0100 kg)•(9.8 m/s/s)  F_{tens} = m•a_{y}
0.098 N  F_{tens} = m•a_{y}
The mass of the 0.0100kg object can be substituted into the equation to yield equation 2.
0.098 N  F_{tens} = (0.0100 kg)•a_{y}
Equations 1 and 2 include an acceleration term  a_{x} and a_{y}. These acceleration values are the same since the system accelerates together; thus, a_{x }= a_{y} = a. An inspection of these two equations show that there are now two equations and two unknowns  F_{tens} and a. These unknown values can be solved for in the customary manner. First, equation 2 is rearranged so as to express F_{tens} in terms of a. This yields equation 3.
F_{tens} = 0.098 N  (0.0100 kg)•a_{y}
Equations 1 and 3 both express F_{tens} in terms of a. The right side of each equation is equal to F_{tens} ; so these two right sides can be set equal to each other to yield the following equation.
(0.5000 kg)•a_{x} = 0.098 N  (0.0100 kg)•a_{y}
Since a_{x} is equal to a_{y}, the equation can be rewritten as
(0.5000 kg)•a = 0.098 N  (0.0100 kg)•a
where a is the horizontal and the vertical acceleration of the 500.0gram and 10.0gram masses. Now the above equation can be manipulated in order to solve for the value of a.
(0.5000 kg)•a + (0.0100 kg)•a = 0.098 N
(0.510 kg)•a = 0.098 N
a = (0.098 N) / (0.510 kg)
a = 0.1922 m/s/s
Now that an acceleration has been determined, the time can be calculated using a kinematic equation. The three known kinematic quantities are:
v_{i} = 0 m/s
a = 0.1922 m/s/s
d = 1.50 m
The following kinematic equation can be used to solve for time (t). The substitutions and algebra are shown.
d = v_{i}•t + 0.5 •a• t^{2}
1.50 m = (0 m/s) • t + 0.5•(.1922 m/s/s)•t^{2}
1.50 m = (0.0961 m/s/s)•t^{2}
t^{2} = (1.50 m) / (0.0961 m/s/s)
t^{2} = 15.6087 s^{2}
t = SQRT(15.6087 s^{2})
t = 3.95 s

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