## Solving Problems with Newton’s Laws

6.1 Solving Problems with Newton’s Laws

### Learning Objectives

By the end of this section, you will be able to:

- Apply problem-solving techniques to solve for quantities in more complex systems of forces
- Use concepts from kinematics to solve problems using Newton’s laws of motion
- Solve more complex equilibrium problems
- Solve more complex acceleration problems
- Apply calculus to more advanced dynamics problems

### Tension Force

**Discuss!**

Consider how you would answer these questions. Then bring this to class for a group discussion.

Two blocks, A and B, are attached by a light string and rest on a smooth surface. The mass of A is 4 kg and the mass of B is 7 kg. Block A is pushed with a force of 12 N and block B is pulled with a force of 26 N.

(a) What is the net force for the entire system (A + B)?

(b) What is the acceleration for the entire system?

(c) What is the tension force in the string that connects the two blocks?

**Practice!**

Practice 6.1.1 |
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Man A (70 kg) and Man B (90 kg) are hanging motionless from a roof. What is the tension, T _{A}, in the top rope? (Assume the ropes are massless and use g = 10 m/s^{2}.) |

(a) 0 N |

(b) 200 N |

(c) 700 N |

(d) 900 N |

(e) 1600 N |

(f) None of the above |

Practice 6.1.2 |
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If you cut the rope between Man A and Man B so that Man A stays motionless, what is the tension, T _{A}, in the top rope? (Assume the ropes are massless and use g = 10 m/s^{2}. Ignore any oscillations resulting from cutting the rope.) |

(a) 0 N |

(b) 200 N |

(c) 700 N |

(d) 900 N |

(e) 1600 N |

(f) None of the above |

Practice 6.1.3 |
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Man A (70kg) and Man B (90kg) are hanging motionless from a platform at rest (assuming we’ve re-attached man B’s body). What is the tension, T _{A}, in the top rope if the platform accelerates upward at a constant rate of 2 m/s^{2}? (Assume the ropes are massless and use g = 10 m/s^{2}.) |

(a) 0 N |

(b) 200 N |

(c) 700 N |

(d) 840 N |

(e) 900 N |

(f) 1600 N |

(g) 1740 N |

(h) 1920 N |

Practice 6.1.4 |
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Before takeoff on an airplane, an inquisitive student on the plane dangles their phone by its earphone wire. It hangs straight down as the plane is at rest waiting to take off. The plane then gains speed rapidly as it moves down the runway. Relative to the student’s hand, the phone will: |

(a) shift toward the front of the plane. |

(b) continue to hang straight down. |

(c) shift toward the back of the plane. |

Practice 6.1.5 |
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The speed of the plane increases at a constant rate over a time interval of several seconds. During this interval, the angle the earphone wire makes with the vertical will: |

(a) increase. |

(b) stay constant. |

(c) decrease. |

**Discuss!**

Consider how you would answer these questions. Then bring this to class for a group discussion.

A 1-kg block is suspended by two ropes, as shown. The rope on the right makes an angle of 26° with the horizontal and pulls on the block with a 4-N tension force. The rope on the left makes an unknown angle of θ with the horizontal and pulls on the block with an unknown amount of tension. What is the force of tension in the left-hand rope?

### Normal Force

**Practice!**

Practice 6.1.6 |
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When a force is called a “normal” force, it is: |

(a) the usual force expected given the arrangement of a system. |

(b) a force that is perpendicular to the surface of the Earth at any given location. |

(c) a contact force perpendicular to the contact surfaces between two solid objects. |

(d) a force that is always vertical. |

(e) the net force acting on a system. |

Practice 6.1.7 |
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A mass m is accelerated down along a frictionless inclined plane. The magnitudes of the forces on the free-body diagram have not been drawn carefully, but the directions of the forces are correct. Which statement below must be true? |

(a) mg > N |

(b) mg < N |

(c) mg = N |

Practice 6.1.8 |
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A mass m is pulled along a frictionless table by a constant external force F_{ext} at some angle above the horizontal. (The magnitudes of the forces on the free-body diagram have not been drawn carefully, but the directions are correct.) Which statement below must be true? |

(a) mg > N |

(b) mg < N |

(c) mg = N |

### Ideal Pulley

**Practice!**

Practice 6.1.9 |
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How does the force exerted on block A by the string (Tension) compare with the weight of block B? (Assume all surfaces are frictionless) |

(a) m_{B}g > T |

(b) m_{B}g < T |

(c) m_{B}g = T |