Newton’s laws of motion – problems and solutions

Newton’s laws of motion – problems and solutions

Newton’s first law of motion

1. If no net force acts on an object, then :

(1) the object is not accelerated

(2) object at rest

(3) the change of velocity of an object = 0

(4) the object can not travels at a constant velocity

Which statement is correct.

Solution

The correct statement :

(1) The object is not accelerated

The net force causes acceleration of an object. So if no net force then objects is not accelerated.

(2) Object at rest

Newton’s first law of motion states that if no net force acts on an object then an object always at rest or the object is always traveling at a constant velocity.

(3) The change of velocity of an object = 0

Change of velocity = acceleration. No change of velocity means no acceleration. If no acceleration then no net force acts on an object.

The object in an elevator

2. The weight of a person in an elevator at rest = 500 N. Acceleration due to gravity is 10 m/s². When lift accelerated, the tension force is 750 N. What is the acceleration of lift.

Known :

Person’s weight (w) = 500 Newton = 500 kg m s^–2 (lift at rest)

Acceleration of gravity (g) = 10 m s^–2

Person’s mass (m) = 500 / 10 = 50 kg

Tension force (T) = 750 N (lift accelerated)

Elevator’s mass ignored.

Wanted: Acceleration of elevator

Solution :

Elevator at rest, no acceleration (a = 0). Force acts upward has plus sign and force acts downward has minus sign.

ΣF = m a

T – w = 0

T = w

T = 500 Newton

If the elevator accelerated downward then the tension force smallest then 500 N. Otherwise, if the elevator accelerated upward then the tension force larger then 500 N.

The tension force = 750 N because the elevator accelerated upward. Force acts upward has plus sign and force acts downward has minus sign.

T – w = m a

750 – 500 = 50 a

250 = 50 a

a = 250 / 50

a = 5.0 m s^–2

3. An 60-kg person in an elevator accelerated downward at 3 m/s². If acceleration due to gravity is 10 m/s², what is the normal force exerted by elevator’s floor on person.

Known :

Mass (m) = 60 kg

Acceleration of person and elevator (a) = 3 m/s²

Acceleration due to gravity (g) = 10 m/s²

Weight (w) = m g = (60)(10) = 600 Newton

Wanted: The normal force (N)

Solution :

There are two forces acts on the person in the elevator, that is weight (w) of person and the normal force (N) exerted by the floor on the person. There are three vector quantities, that is weight (w), normal force (N) and acceleration of elevator, where weight acts downward, the normal force acts upward, acceleration of elevator is downward. Vector quantities that act downward have plus sign and vector quantities that act upward have minus sign.

∑F = m a

w – N = (60)(3)

600 – N = 180

N = 600 – 180

N = 420 Newton

4. A 40-kg object in an elevator accelerated upward. If the elevator’s floor exerts 520 N on object and acceleration due to gravity is 10 m/s². What is the acceleration of the elevator?

Known :

Mass (m) = 40 kg

Normal force (N) = 520 N

Acceleration due to gravity (g) = 10 m/s²

weight (w) = m g = (40)(10) = 400 N

Wanted : Acceleration of elevator

Solution :

∑F = m a

400 – 520 = (40)(a)

-120 = (40)(a)

a = -120/40

a = -3 m/s²

Acceleration of elevator is 3 m/s². Minus sign indicates that elevator travels upward.

5. An 60-kg object in an elevator accelerated downward at 3 m/s². What is the force exerted by object on the elevator’s floor.

Known :

Mass (m) = 60 kg

Weight (w) = m g = (60 kg)(10 m/s²) = 600 kg m/s² = 600 Newton

Acceleration of elevator (a) = 3 m/s², downward

Wanted : Force exerted by object on the elevator’s floor.

Solution :

Elevator accelerated downward at 3 m/s. Force acts downward has plus sign and force acts upward has minus sign.

w – N = m a

N = w – m a

N = 600 – (60)(3)

N = 600 – 180

N = 420 Newton

Force exerted by the object on the elevator’s floor = 420 N.

6. Two blocks are connected by a cord running over a pulley. Ignore the mass of the cord and pulley and any friction in the pulley. Mass of block A is 6 kg and mass of block B is 2 kg. Acceleration due to gravity is 10 m/s². What is the tension force?

Known :

m_A = 6 kg, m_B = 2 kg, g = 10 m/s²

w_A = m_A g = (6 kg)(10 m/s²) = 60 kg m/s²

w_B = m_B g = (2 kg)(10 m/s²) = 20 kg m/s²

Wanted : tension force (T) ?

Solution :

w_A > w_B so that m_A moves downward, m_B moves upward.

Newton’s second law :

ΣF = m a

w_A – w_B = (m_A + m_B) a

60 – 20 = (6 + 2) a

40 = (8) a

a = 40 / 8 = 5 m/s²

Tension force :

m_A moves downward :

w_A – T_A = m_A a

60 – T_A = (6)(5)

60 – T_A = 30

T_A = 60 – 30

T₂ = 30 Newton

m_B moves upward :

T_B – w_B = m_B a

T_B – 20 = (2)(5)

T_B – 20 = 10

T_B = 10 + 20

T₁ = 30 Newton

Tension force (T) = 30 Newton.

7. Mass of object A = 5 kg, acceleration due to gravity (g) = 10 m s^-2. Object A moves downward at 2.5 m.s^-2. What is the mass of B ?

Known :

Mass A (m_A) = 5 kg

Acceleration due to gravity (g) = 10 m/s²

Acceleration of object A (a) = 2.5 m/s²

Weight A (w_A) = (m_A)(g) = (5)(10) = 50 Newton

Wanted : Mass of object B (m_B)

Solution :

Block A moves downward so weight of object A (w_A) larger than weight of object B (w_B).

Apply Newton’s second law :

ΣF = m a

w_A – w_B = (m_A + m_B) a

50 – (m_B)(10) = (5 + m_B) (2.5)

50 – 10 m_B = 12.5 + 2.5 m_B

50 – 12.5 = 2.5 m_B + 10 m_B

37.5 = 12.5 m_B

m_B = 3 kg

8. Acceleration due to gravity is 10 m/s². What is the tension force.

Known :

Mass of object 1 (m₁) = 2 kg

Mass of object 2 (m₂) = 3 kg

Acceleration due to gravity (g) = 10 m/s²

Weight 1 (w₁) = (m₁)(g) = (2 kg)(10 m/s²) = 20 kg m/s²

Weight 2 (w₂) = (m₂)(g) = (3 kg)(10 m/s²) = 30 kg m/s²

Wanted : tension force (T)

Solution :

w₂ > w₁ so that m₂ moves downward and m₁ moves upward.

Newton’s second law of motion :

ΣF = m a

w₂ – w₁ = (m₁ + m₂) a

30 – 20 = (2 + 3 ) a

10 = (5) a

a = 10 / 5 = 2 m/s².

Tension force ?

m₂ moves downward

w₂ – T₂ = m₂ a

30 – T₂ = (3)(2)

30 – T₂ = 6

T₂ = 30 – 6

T₂ = 24 Newton

m₁ moves upward

T₁ – w₁ = m₁ a

T₁ – 20 = (2)(2)

T₁ – 20 = 4

T₁ = 20 + 4

T₁ = 24 Newton

Tension force (T) = 24 Newton.

1. Question: Why does Newton’s first law of motion emphasize the importance of external forces in changing an object’s state of motion? Answer: Newton’s first law of motion, often called the law of inertia, states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. This law highlights that it’s the external forces, and not internal qualities of the object itself, that can change its state of motion.
2. Question: How does Newton’s second law relate force, mass, and acceleration? Answer: Newton’s second law of motion states that the force acting on an object is equal to the mass of that object multiplied by its acceleration ($F=ma$). This establishes a direct proportionality between force and acceleration and an inverse proportionality between mass and acceleration, given a constant force.
3. Question: Why don’t you feel the Earth moving even though it’s constantly in motion? Answer: This is an illustration of Newton’s first law. We, along with the Earth and everything on it, are moving uniformly together. Without an external force to change this state of motion, there’s no feeling of acceleration or change, making it seem as though we’re stationary.
4. Question: How does Newton’s third law explain the action of a rocket in space? Answer: Newton’s third law states that for every action, there’s an equal and opposite reaction. A rocket in space propels itself by expelling exhaust gases backward. The action is the gas moving backward, and the reaction is the rocket moving forward.
5. Question: Why do you move forward in a car when it suddenly stops? Answer: According to Newton’s first law, your body continues its state of motion (forward) when the car stops. Unless a force (like a seatbelt) acts on you, you’ll continue moving forward due to inertia.
6. Question: If a feather and a hammer are dropped in a vacuum, what can Newton’s laws predict about their motion? Answer: In a vacuum, there’s no air resistance. Based on Newton’s first and second laws, both the feather and the hammer should fall at the same rate and hit the ground simultaneously because the only force acting on them is gravity.
7. Question: How does an airbag reduce the chances of injury during a car crash? Answer: Newton’s first law indicates that an object in motion stays in motion. In a crash, a person’s forward motion continues. The airbag provides a cushioned surface that increases the time of deceleration, thereby reducing the force experienced by the person according to $F=ma$.
8. Question: Why is it easier to push an empty shopping cart than a loaded one? Answer: Newton’s second law ($F=ma$) suggests that for the same amount of force applied, an object with less mass (empty cart) will have a greater acceleration than an object with more mass (loaded cart).
9. Question: Why does a swimmer push water backward to move forward? Answer: This is a demonstration of Newton’s third law. When the swimmer pushes water backward (action), there’s an equal and opposite reaction which propels the swimmer forward.
10. Question: How does friction relate to Newton’s first law? Answer: Friction is an external force that can act on objects to slow them down or stop their motion. Without friction (or any other external force), an object would continue its state of motion indefinitely as stated by Newton’s first law.