Force: Problem Set with Solutions

This problem set is designed to accompany the PhysicsLibrary entry force. The exercises are stated first for self-study. Complete, worked solutions follow afterward.

Exercises

Problem 1. Net force from perpendicular forces. Figure 1 shows a ring pulled by two perpendicular forces, $\mathbf F_1=6\,\mathrm{N}$ to the right and $\mathbf F_2=8\,\mathrm{N}$ upward. Determine the magnitude and direction of the net force.

Figure 1: Two perpendicular forces acting on a ring.

Problem 2. Horizontal motion with kinetic friction. A $4.0\,\mathrm{kg}$ block is pushed across a horizontal floor by a constant horizontal force of $25\,\mathrm{N}$ , as shown in Figure 2. The coefficient of kinetic friction is $\mu_k=0.30$ . Find (a) the normal force, (b) the kinetic friction force, and (c) the horizontal acceleration of the block.

Figure 2: Free-body diagram for a block pushed on a rough horizontal surface.

Problem 3. Static friction on an incline. A $5.0\,\mathrm{kg}$ block rests on a $25^\circ$ incline, as in Figure 3. Assume the block is just on the verge of slipping downward. Find the minimum coefficient of static friction required to keep it at rest.

Figure 3: Block on an incline with weight, normal force, and static friction.

Problem 4. An ideal Atwood machine. In the system shown in Figure 4, $m_1=2.0\,\mathrm{kg}$ and $m_2=3.0\,\mathrm{kg}$ . The string and pulley are ideal. Find (a) the magnitude of the acceleration and (b) the string Tension.

Figure 4: Free-body diagram for an ideal Atwood machine.

Problem 5. Spring force and simple horizontal motion. A $0.40\,\mathrm{kg}$ block is attached to a horizontal spring with spring constant $k=80\,\mathrm{N/m}$ . The spring is stretched by $x=0.15\,\mathrm{m}$ and released from rest, as in Figure 5. Find (a) the magnitude of the initial spring force and (b) the initial acceleration of the block.

Figure 5: Block attached to a stretched horizontal spring.

Problem 6. Centripetal force on a flat curve. A $1200\,\mathrm{kg}$ CAR travels at $12\,\mathrm{m/s}$ around a flat circular curve of radius $50\,\mathrm{m}$ , as sketched in Figure 6. Find (a) the required centripetal force and (b) the minimum coefficient of static friction required if static friction provides the centripetal force.

Figure 6: Car undergoing uniform circular motion on a flat curve.

Worked solutions

Solution 1. Because the two forces are perpendicular, the magnitude of the net force is found from the Pythagorean theorem:

$\displaystyle F_{\rm net}=\sqrt{F_1^2+F_2^2} =\sqrt{6^2+8^2} =10\,\mathrm{N}.$

The direction above the positive horizontal axis is

$\displaystyle \theta=\tan^{-1}\left(\frac{8}{6}\right)\approx 53.1^\circ.$

Therefore the net force is

$\displaystyle \boxed{F_{\rm net}=10\,\mathrm{N},\qquad \theta\approx 53.1^\circ\text{ above the horizontal}.}$

Solution 2. On a horizontal surface with no vertical acceleration,

$\displaystyle N=mg=(4.0)(9.8)=39.2\,\mathrm{N}.$

The kinetic friction force is

$\displaystyle f_k=\mu_kN=(0.30)(39.2)=11.76\,\mathrm{N}\approx 11.8\,\mathrm{N}.$

The net horizontal force is

$\displaystyle F_{\rm net,x}=F_{\rm push}-f_k=25.0-11.76=13.24\,\mathrm{N}.$

Thus the acceleration is

$\displaystyle a=\frac{F_{\rm net,x}}{m}=\frac{13.24}{4.0}=3.31\,\mathrm{m/s^2}.$

So the answers are

$\displaystyle \boxed{N=39.2\,\mathrm{N},\qquad f_k\approx 11.8\,\mathrm{N},\qquad a\approx 3.31\,\mathrm{m/s^2}.}$

Solution 3. Resolve the weight into components parallel and perpendicular to the incline:

$\displaystyle mg\sin\theta$ down the incline $\displaystyle , \qquad mg\cos\theta$ into the incline $\displaystyle .$

At the threshold of slipping, static friction takes its maximum value,

$\displaystyle f_{s,\max}=\mu_sN.$

Since the block is in equilibrium,

$\displaystyle mg\sin\theta=f_{s,\max}=\mu_sN, \qquad N=mg\cos\theta.$

Therefore

$\displaystyle \mu_s=\frac{mg\sin\theta}{mg\cos\theta}=\tan\theta=\tan 25^\circ\approx 0.466.$

Hence the minimum coefficient is

$\displaystyle \boxed{\mu_s^{\min}\approx 0.47.}$

Solution 4. Because $m_2>m_1$ , mass $m_2$ moves downward and $m_1$ moves upward. Applying Newton's second law to each mass gives

$\displaystyle T-m_1g=m_1a,$

$\displaystyle m_2g-T=m_2a.$

Adding the equations,

$\displaystyle (m_2-m_1)g=(m_1+m_2)a,$

$\displaystyle a=\frac{(m_2-m_1)g}{m_1+m_2} =\frac{(3.0-2.0)(9.8)}{3.0+2.0} =1.96\,\mathrm{m/s^2}.$

Now solve for the tension:

$\displaystyle T=m_1(g+a)=2.0(9.8+1.96)=23.52\,\mathrm{N}.$

Therefore

$\displaystyle \boxed{a=1.96\,\mathrm{m/s^2},\qquad T\approx 23.5\,\mathrm{N}.}$

Solution 5. Hooke's law gives the magnitude of the spring force:

$\displaystyle F_s=kx=(80)(0.15)=12\,\mathrm{N}.$

Initially this is the net horizontal force on the block, so

$\displaystyle a=\frac{F_s}{m}=\frac{12}{0.40}=30\,\mathrm{m/s^2}.$

The spring force and acceleration are directed toward equilibrium. Thus

$\displaystyle \boxed{F_s=12\,\mathrm{N},\qquad a=30\,\mathrm{m/s^2}.}$

Solution 6. The required centripetal force is

$\displaystyle F_c=\frac{mv^2}{r} =\frac{(1200)(12^2)}{50} =3456\,\mathrm{N}.$

On a flat curve, the centripetal force is supplied by static friction. The normal force is

$\displaystyle N=mg=(1200)(9.8)=11760\,\mathrm{N}.$

Thus the minimum coefficient of static friction is

$\displaystyle \mu_s^{\min}=\frac{F_c}{N}=\frac{3456}{11760}\approx 0.294.$

Therefore

$\displaystyle \boxed{F_c=3456\,\mathrm{N},\qquad \mu_s^{\min}\approx 0.29.}$

Study notes

These six problems illustrate several core ideas related to force:

forces add vectorially;
free-body diagrams help isolate all forces acting on a body;
friction forces must be modeled carefully, distinguishing static and kinetic friction;
Newton's second law connects force to acceleration;
spring forces are modeled by Hooke's law;
centripetal force is not a new interaction but the inward net force needed for circular motion.

Bibliography

1: Daniel Kleppner and Robert J. Kolenkow, An Introduction to Mechanics, 2nd ed., Cambridge University Press, 2014.
2: John R. Taylor, Classical Mechanics, University Science Books, 2005.
3: David Halliday, Robert Resnick, and Jearl Walker, Fundamentals of Physics, 10th ed., Wiley, 2013.

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