Applications Of Newton's First and Second Laws

Helena Dedic

Beware: Many of the solutions to these exercises use $g=10m/s^{2}$ !

Exercise 1

A 40 kg child is riding in a hot air balloon which is descending at a speed of 1 m/s. What is the normal force exerted on the child?

• SOLUTION EX 1
  
  

Exercise 2

Two blocks, $m_{1}=10$ kg and $m_{2}=5$ kg, connected as shown in the diagram, move at constant velocity towards the left. Find the normal force on both blocks and the tension in the rope given that the force of friction on the block sliding horizontally is 15 N.

• SOLUTION EX 2
  
  

Exercise 3

Each block shown below has a mass of 10 kg. Find the magnitude and direction of the normal force acting on each block.

• SOLUTION EX 3
  
  

Exercise 4

Two blocks, $m_{1}=4$ kg and $m_{2}=8$ kg are connected as shown in the diagram and $m_{1}$ moves at constant velocity up the incline because $m_{2}$ is pulled by a force F. Find the force F exerted on the block sliding along the horizontal towards the left, the normal force on both blocks and the tension in the rope given that the force of friction on both blocks is 20 N.

• SOLUTION EX 4
  
  

Exercise 5

Sally holds the two boxes, $m_{1}=18$ kg and $m_{2}=25$ kg at rest on the incline by pushing the blue box with a force F. Draw a free body diagram for each of the two boxes. Assume that there is no friction exerted on either of the two boxes. Determine the normal forces exerted on each of the boxes and the force exerted by Sally.

• SOLUTION EX 5
  
  

Exercise 6

A 20 kg child starts from rest and travels 3 m down a slide that is inclined 35° to the horizontal. If her speed at the bottom is 1 m/s, what was the frictional force along the slide?

• SOLUTION EX 6
  
  

Exercise 7

A man pushes a 20 kg lawn mover with a force of 80 N directed along the handle, which is inclined at $30^{o}$ to the horizontal.

(a) If he moves at constant velocity, what is the retarding force due to the ground?

(b) What force along the handle would produce an acceleration of 1 m/$s^{2}$ given the same retarding force?

• SOLUTION EX 7
  
  

Exercise 8

A child drops from a ledge 1 m above the floor. Estimate the force on her 40 kg torso when she lands by bending her knees and stopping her torso in 30 cm or lands stiffly and her torso is stopped within 4 cm.

• SOLUTION EX 8
  
  

Exercise 9

Two blocks with masses $m_{1}=5$ kg and $m_{2}=3$ kg are connected by a light rope. The mass $m_{1}$ is placed on a frictionless incline that makes 30° with the horizontal. A force $F_{0}=10$ N acts on $m_{2}$ at 20° to the horizontal. Find the acceleration of the system and the tension in the rope.

• SOLUTION EX 9
  
  

Exercise 10

Would it be easier to play catch with a large ball in the outer space since it has no weight?

• SOLUTION EX 10
  
  

Exercise 11

What is the reading on the spring scale for each of the situations depicted? Each block has a mass of 5 kg.

a) b) c)

• SOLUTION EX 11
  
  

Exercise 12

A ball thrown vertically upward momentarily comes to rest at its highest point. Is it in equilibrium at this instant? Explain why or why not.

• SOLUTION EX 12
  
  

Exercise 13

The 50 kg torso of a sprinter starts at rest and reaches 6 m/s within 80 cm, both measured horizontally. Estimate the acceleration and the horizontal force exerted on the torso-hip joint.

• SOLUTION EX 13
  
  

Exercise 14

A length of rope has a mass of 30 g. It is used to accelerate a 200 g object vertically upward at $4m/s^{2}$. What is the tension in the rope - at the bottom of the rope, at the top of the rope and in the middle of the rope?

• SOLUTION EX 14
  
  

Exercise 15

A block of mass 1 kg is on a frictionless $37^{o}$ incline and is subject to a horizontal force of 5 N. What is its acceleration? If it is initially moving up the incline at 4 m/s, what is its displacement along the incline in 2 s.

• SOLUTION EX 15
  
  

Exercise 16

A 3 kg block is suspended with two ropes, one of which is horizontal. Find the tension in each rope.

• SOLUTION EX 16
  
  

Exercise 17

A car is moving at 90 km/h. A 75-kg driver is held firmly in the seat by a seatbelt. Find the average force exerted on the driver by the seatbelt (in other words, assume that the force is constant) if the car crashes by crumpling through a distance of 75 cm.

• SOLUTION EX 17
  
  

Exercise 18

Two blocks with masses $m_{a}=0.2$ kg and $m_{b}=0.3$ kg hang under one another. Find the tensions in the (massless) ropes in the following situations:

(a) the blocks are at rest;

(b) they move upward at 5 m/s;

(c) they accelerate upward at 2 m/$s^{2}$;

(d) they accelerate downward at 2 m/$s^{2}$.

(e) if the maximum allowable tension in the rope is 10 N, what is the maximum possible upward acceleration?

• SOLUTION EX 18
  
  

Exercise 19

Two blocks are connected by a massless rope. The horizontal surface is frictionless. If $m_{1}=2$ kg for what value of $m_{2}$ will the

(a) acceleration of the system be 4 m/$s^{2}$ or

(b) the tension in the rope be 8 N?

• SOLUTION EX 19
  
  

Exercise 20

A body experiences a force of 10 N directed at $30^{o}$ N of E and a force of 12.8 N directed at $20^{o}$ E of S. What third force would keep the body at equilibrium?

• SOLUTION EX 20
  
  

Exercise 21

What is the spring constant of a spring which stretches 1.5 cm when supporting a 2 kg mass on a $37^{o}$ slope?

• SOLUTION EX 21
  
  

Exercise 22

A 70 kg pilot flies an airplane at 400 km/h (55.6 m/s) in a turn of radius 2 km ($2\times 10^{3}$ m).

(a) At what angle $\theta$ to the horizontal are the wings?

(b) What is the normal force exerted on the pilot (we often refer to it as pilot's apparent weight)?

• SOLUTION EX 22
  
  

Exercise 23

A 60-kg woman is on a Ferris wheel that takes her in a vertical circle of radius 20 m at constant speed.

(a) At what speed would she feel weightless (the normal force is equal to zero) at the top?

(b) At this speed, what would be the normal force exerted on her (her apparent weight) at the bottom?

• SOLUTION EX 23
  
  

Exercise 24

A doughnut-shaped spacestation travels in outer space where the gravitational force is negligible. In order to generate "a feel of weight", the spacestation is rotating with constant speed v. It has a diameter of 2 km (radius $r=10^{3}$ m). An astronaut walking in the spacestation is shown in the diagram below. What should be the period of rotation for the astronaut to experience an apparent weight of 20% that on Earth?

• SOLUTION EX 24
  
  

Exercise 25

A spring stretches 1.5 cm when it supports a 200 g mass. How much does it stretch when it supports a 250 g mass?

• SOLUTION EX 25
  
  

Exercise 26

A pendulum bob is held in place by a spring as shown. If the spring constant $k=3.92\times 10^{3}$ N/m, and the spring stretches 2 cm, what is the mass of the bob?

• SOLUTION EX 26
  
  

Exercise 27

A spring of spring constant $k=2\times 10^{3}$ N/m supports a mass of 2 kg.

(a) How much does the spring stretch when the spring moves up at constant velocity?

(b) How much does the spring stretch when the spring moves up with an acceleration of $2m/s^{2}$?

(c) What is the acceleration if the spring stretches 0.8 cm?

• SOLUTION EX 27
  
  

Exercise 28

A 40 kg child is riding in a hot air balloon which is descending with an acceleration of $1m/s^{2}$. What is the apparent weight of the child?

• SOLUTION EX 28
  
  

Exercise 29

It is possible to twirl a bucket containing water in a vertical circle without the water spilling out.

(a) What are the forces acting on the water at the highest point?

(b) Why doesn't the water spill out?

• SOLUTION EX 29
  
  

Exercise 30

Two blocks, $m_{1}=3$ kg and $m_{2}=5$ kg are connected as shown in the diagram and move at constant velocity down the incline. Find the coefficient of kinetic friction assuming it is the same for both blocks and the tension in the rope.

• SOLUTION EX 30
  
  

Exercise 31

A 5 kg block is on a 37º incline for which $\mu _{k}$ = 0.1. It is acted on by a horizontal force of 25 N. What is the acceleration of the block if it is moving up the incline? If its initial speed is 6 m/s up the incline, how far does it travel in 2 s?

• SOLUTION EX 31
  
  

Exercise 32

The minimum stopping distance for a car travelling at an initial speed of 100 km/h is 60 m on level ground. What is the stopping distance when it moves down a 10º incline, or up a 10º incline? Assume the same initial speed and surface.

• SOLUTION EX 32
  
  

Exercise 33

A circular off ramp has a radius of 60 m and a posted speed limit of 60 km/h. If the road is horizontal, what is the minimum coefficient of friction required?

• SOLUTION EX 33
  
  

Exercise 34

A 5 kg block is subject to a horizontal force of 30 N. It lies on a surface for which $\mu _{k}$ = 0.5 and $\mu _{s}$ = 0.7. If the block is at rest, what is the frictional force on it? What is the acceleration of the block if it is moving to the left or to the right?

• SOLUTION EX 34
  
  

Exercise 35

The moon Io of Jupiter is in a circular orbit of radius $4.22\times 10^{5}$ km with a period of 1.77 days. The period of another moon, Europa, is 3.55 days. What is the radius of its orbit? What is the mass of Jupiter?

• SOLUTION EX 35
  
  

Exercise 36

In November 1984, the space shuttle Discovery was placed in a circular orbit at an altitude of 315 km in order to catch up to the disabled Westar 6 satellite in circular orbit at an altitude of 360 km. Suppose that these two objects were initially on opposite sides of the earth. How many orbits would the satellite have made for the shuttle to be beneath the satellite (that is along a radial line and closer to Earth)? Take $r_{earth}$ = 6370 km.

• SOLUTION EX 36
  
  

Exercise 37

A particle of mass 4M is at the origin while a particle of mass M is located at x = 1 m. A third particle of mass m is located somewhere in the vicinity of the other two particles. Assume that gravitational forces between particles are the only interactions involved in this problem (in other words, assume that these three particles are located somewhere in outer space.) Where should the particle of mass m be placed so that it is in equilibrium?

• SOLUTION EX 37
  
  

Exercise 38

Calculate the strength of the gravitational field at the surface of the following bodies: Neptune ($m=1.03\times 10^{26}$ kg, $r=2.43\times 10^{7}$ m), Jupiter ($m=1.9\times 10^{27}$ kg, $r=7.14\times 10^{7}$ m), and a neutron star ($m=10^{30}$ kg, $r=20$ km). In all cases, assume that the objects are spheres of uniform density. Assume that the centre of mass is at the centre of the sphere in all cases.

• SOLUTION EX 38
  
  

Exercise 39

Estimate the magnitude of the force exerted on the moon by the Earth and by the Sun.

• SOLUTION EX 39
  
  

Exercise 40

At what point between the Earth and the Moon does the net force on a spacecraft due to these two bodies vanish? Assume the mass of the Earth is 81 times that of the Moon.

• SOLUTION EX 40
  
  

Exercise 41

Use the moon's period of 27.32 days and the radius of its orbit $3.84\times 10^{5}$ km to find the mass of the Earth. Use the Earth-Sun distance and the period of the Earth to find the mass of the Sun.

• SOLUTION EX 41
  
  

Exercise 42

Neutron stars rotate with periods less than one second. The typical radii of neutron stars are just a few kilometers. Let assume that a neutron star rotates with a period of 1 s and has the radius of 20 km. We can estimate the mass of this star from the fact that objects on its surface are bound to remain on it. Determine the mass of this star.

• SOLUTION EX 42
  
  

Exercise 43

A 70 kg man is shown enjoying the "rotor" ride at the amusement park. The rotor has a diameter of 4 m and rotates once every 2 s. What is the normal force exerted on the man? What is the force of friction required to keep him from slipping down the wall?

• SOLUTION EX 43
  
  

Exercise 44

Two blocks with equal masses $m_{1}=m_{2}=5$ kg are connected via a pulley. The coefficient of kinetic friction is $\mu _{k}$ = 0.25.

(a) Find the acceleration of the two blocks when $m_{1}$ is moving down.

(b) Find the acceleration of the two blocks when $m_{1}$ is moving up.

(c) Determine the mass of $m_{1}$ so that it moves up at constant speed when $m_{2}=6$ kg.

• SOLUTION EX 44