Physics 170 Homework 4: Friction, Inclined Planes, Circular Motion, and Centripetal Force

Q10. A wet bar of soap slides freely down a ramp 9.0 m long inclined at 8.0°. How long does it take to reach the bottom? Assume μk = 0.060.

Background

Topic: Dynamics on an Inclined Plane with Friction

This question tests your understanding of forces on an inclined plane, including gravity, normal force, and kinetic friction, and how to use Newton's second law to find acceleration and time.

Key Terms and Formulas

• Kinetic friction: $f_k = \mu_k N$

• Newton's second law: $F_{\text{net}} = ma$

• Component of gravity down the incline: $mg \sin \theta$

• Normal force: $N = mg \cos \theta$

• Kinematic equation for distance: $d = \frac{1}{2} a t^2$ (starting from rest)

Step-by-Step Guidance

1. Draw a free-body diagram for the soap on the incline. Identify all forces: gravity (split into parallel and perpendicular components), normal force, and kinetic friction.

2. Write the equation for the net force parallel to the incline: $F_{\text{net}} = mg \sin \theta - f_k$.

3. Express the friction force in terms of the normal force: $f_k = \mu_k N = \mu_k mg \cos \theta$.

4. Combine the above to find the net force and then solve for the acceleration $a$ down the ramp.

5. Set up the kinematic equation $d = \frac{1}{2} a t^2$ to solve for the time $t$ it takes to reach the bottom, but do not solve for $t$ yet.

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Q14. Police investigators, examining the scene of an accident involving two cars, measure 72-m-long skid marks of one of the cars, which nearly came to a stop before colliding. The coefficient of kinetic friction between rubber and the pavement is about 0.80. Estimate the initial speed of that car assuming a level road.

Background

Topic: Work-Energy Principle with Friction

This question tests your ability to relate the work done by friction to the change in kinetic energy, and to solve for the initial speed of a car based on the length of skid marks.

Key Terms and Formulas

• Kinetic friction: $f_k = \mu_k N$

• Work done by friction: $W = f_k d$

• Kinetic energy: $KE = \frac{1}{2} m v^2$

• Work-Energy Theorem: $W_{\text{friction}} = \Delta KE$

Step-by-Step Guidance

1. Write the work-energy theorem: the work done by friction equals the change in kinetic energy as the car comes to rest.

2. Express the work done by friction: $W = -f_k d = -\mu_k m g d$ (negative because friction opposes motion).

3. Set the change in kinetic energy: $\Delta KE = 0 - \frac{1}{2} m v_0^2$ (final speed is zero).

4. Set $W = \Delta KE$ and solve for the initial speed $v_0$ in terms of the given quantities, but do not calculate $v_0$ yet.

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Q23. In Fig. 5-35 the coefficient of static friction between mass mA and the table is 0.40, whereas the coefficient of kinetic friction is 0.30. (a) What minimum value of mA will keep the system from starting to move? (b) What value(s) of mA will keep the system moving at constant speed?

Background

Topic: Static and Kinetic Friction in Connected Systems

This question tests your understanding of frictional forces, equilibrium, and Newton's laws for systems with pulleys and connected masses.

Key Terms and Formulas

• Static friction: $f_s = \mu_s N$

• Kinetic friction: $f_k = \mu_k N$

• Newton's second law: $F_{\text{net}} = ma$

• Equilibrium: $F_{\text{net}} = 0$ (for constant speed or no motion)

Step-by-Step Guidance

1. Draw a free-body diagram for each mass. Identify all forces acting on $m_A$ (tension, friction, normal force, gravity) and $m_B$ (tension, gravity).

2. For part (a), set up the equilibrium condition for $m_A$ just before motion starts: tension equals maximum static friction.

3. Express the tension in terms of $m_B$ and $g$ (since $m_B$ hangs vertically).

4. Set up the equation $m_B g = \mu_s m_A g$ and solve for the minimum $m_A$ that prevents motion, but do not calculate $m_A$ yet.

5. For part (b), repeat the process using kinetic friction for constant speed, and set up the equation for $m_A$.

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Q28. Two masses mA = 2.0 kg and mB = 5.0 kg are on inclines and are connected together by a string as shown in Fig. 5-37. The coefficient of kinetic friction between each mass and its incline is μk = 0.30. If mA moves up, and mB moves down, determine their acceleration.

Background

Topic: Dynamics of Connected Masses on Inclined Planes with Friction

This question tests your ability to analyze forces on inclined planes, including friction, and to apply Newton's second law to a system of connected objects.

Key Terms and Formulas

• Kinetic friction: $f_k = \mu_k N$

• Newton's second law for each mass: $F_{\text{net}} = ma$

• Components of gravity: $mg \sin \theta$ (parallel), $mg \cos \theta$ (perpendicular)

Step-by-Step Guidance

1. Draw free-body diagrams for both $m_A$ and $m_B$, showing all forces (gravity components, friction, tension).

2. Write Newton's second law for each mass along the direction of motion, including friction and the incline angle for each.

3. Express the normal force for each mass: $N_A = m_A g \cos \theta_A$, $N_B = m_B g \cos \theta_B$.

4. Set up the equations for the net force on each mass, and combine them to solve for the acceleration $a$ of the system, but do not calculate $a$ yet.

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Q29. A child slides down a slide with a 34° incline, and at the bottom her speed is precisely half what it would have been if the slide had been frictionless. Calculate the coefficient of kinetic friction between the slide and the child.

Background

Topic: Energy Conservation with Friction

This question tests your understanding of energy conservation, the effect of friction on mechanical energy, and how to relate speed to energy loss due to friction.

Key Terms and Formulas

• Potential energy lost: $\Delta PE = mgh$

• Kinetic energy at the bottom: $KE = \frac{1}{2} m v^2$

• Work done by friction: $W_{\text{friction}} = f_k d$

• Kinetic friction: $f_k = \mu_k N$

• Energy conservation: $\Delta PE = KE + W_{\text{friction}}$

Step-by-Step Guidance

1. Write the energy conservation equation, including the work done by friction.

2. Express the work done by friction in terms of $\mu_k$, the normal force, and the distance along the slide.

3. Relate the final speed to the frictionless case (i.e., $v = \frac{1}{2} v_0$ where $v_0$ is the speed with no friction).

4. Set up the equation to solve for $\mu_k$, but do not calculate $\mu_k$ yet.

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Q42. How large must the coefficient of static friction be between the tires and the road if a car is to round a level curve of radius 85 m at a speed of 95 km/h?

Background

Topic: Circular Motion and Static Friction

This question tests your understanding of centripetal force, static friction, and how to relate speed, radius, and friction for a car rounding a curve.

Key Terms and Formulas

• Centripetal force: $F_c = \frac{mv^2}{r}$

• Static friction: $f_s = \mu_s N$

• Normal force on level ground: $N = mg$

Step-by-Step Guidance

1. Write the equation for the maximum static friction force: $f_{s,\text{max}} = \mu_s mg$.

2. Set the required centripetal force equal to the maximum static friction: $\mu_s mg = \frac{mv^2}{r}$.

3. Cancel mass $m$ from both sides and solve for $\mu_s$ in terms of $v$ and $r$.

4. Convert the speed from km/h to m/s before plugging into the equation, but do not calculate $\mu_s$ yet.

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Q43. Suppose the space shuttle is in orbit 400 km from the Earth's surface, and circles the Earth about once every 90 min. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g, the gravitational acceleration at the Earth's surface.

Background

Topic: Circular Motion and Centripetal Acceleration in Orbit

This question tests your understanding of orbital motion, centripetal acceleration, and how to relate orbital period and radius to acceleration.

Key Terms and Formulas

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

• Orbital speed: $v = \frac{2\pi r}{T}$

• Radius of orbit: $r = R_{\text{Earth}} + 400,000$ m

• Period: $T = 90$ min $= 5400$ s

Step-by-Step Guidance

1. Calculate the total radius of the orbit by adding Earth's radius to the altitude above the surface.

2. Express the orbital speed $v$ in terms of $r$ and $T$.

3. Plug $v$ into the centripetal acceleration formula to get $a_c$ in terms of $r$ and $T$.

4. Express $a_c$ as a fraction of $g$ (i.e., $a_c/g$), but do not calculate the final value yet.

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Q44. A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N. (a) Find the speed of the bucket. (b) How fast must the bucket move at the top of the circle so that the rope does not go slack?

Background

Topic: Circular Motion and Tension in a Vertical Circle

This question tests your understanding of forces in vertical circular motion, including tension, gravity, and the conditions for maintaining tension throughout the motion.

Key Terms and Formulas

• Centripetal force: $F_c = \frac{mv^2}{r}$

• At the lowest point: $T - mg = \frac{mv^2}{r}$

• At the top: $T_{\text{min}} + mg = \frac{mv^2}{r}$ (for $T_{\text{min}} = 0$ for just not going slack)

Step-by-Step Guidance

1. For part (a), write the force equation at the lowest point and solve for $v$ in terms of $T$, $m$, $g$, and $r$.

2. For part (b), set the tension at the top to zero and solve for the minimum speed required to keep the rope taut.

3. Set up the equations but do not calculate the speeds yet.

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Q47. A jet pilot takes his aircraft in a vertical loop (Fig. 5-43). (a) If the jet is moving at a speed of 1200 km/h at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 6.0 g's. (b) Calculate the 78-kg pilot’s effective weight (the force with which the seat pushes up on him) at the bottom of the circle, and (c) at the top of the circle (assume the same speed).

Background

Topic: Circular Motion, Centripetal Acceleration, and Apparent Weight

This question tests your understanding of centripetal acceleration, the relationship between speed, radius, and acceleration, and the concept of apparent weight in circular motion.

Key Terms and Formulas

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

• Apparent weight at the bottom: $N = m(g + a_c)$

• Apparent weight at the top: $N = m(a_c - g)$

Step-by-Step Guidance

1. For part (a), set $a_c = 6g$ and solve for the minimum radius $r$ in terms of $v$ and $g$.

2. Convert the speed from km/h to m/s before plugging into the equation.

3. For part (b), calculate the normal force (effective weight) at the bottom using $N = m(g + a_c)$.

4. For part (c), calculate the normal force at the top using $N = m(a_c - g)$, but do not compute the final values yet.

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Q54. Two blocks, with masses mA and mB, are connected to each other and to a central post by cords as shown in Fig. 5-46. They rotate about the post at frequency f (revolutions per second) on a frictionless horizontal surface at distances rA and rB from the post. Derive an algebraic expression for the tension in each segment of the cord (assumed massless).

Background

Topic: Circular Motion and Tension in Rotating Systems

This question tests your ability to analyze forces in circular motion, especially tension in cords connecting rotating masses.

Key Terms and Formulas

• Centripetal force: $F_c = m \omega^2 r$

• Angular velocity: $\omega = 2\pi f$

• Tension provides the required centripetal force for each mass

Step-by-Step Guidance

1. Draw a free-body diagram for each block, showing the tensions acting on them.

2. Write Newton's second law for each block in the radial direction, expressing the net force as the difference in tensions (if applicable) and equate to the required centripetal force.

3. Express the centripetal force in terms of $m$, $r$, and $f$ using $\omega = 2\pi f$.

4. Set up the system of equations to solve for the tension in each segment, but do not solve for the tensions yet.

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