3 Questions about Rope tension equation

1. The picture below shows three blocks, namely A, B and C which are located on a smooth horizontal plane. If mass A = 1 kg, mass B = 2 kg and mass C = 2 kg and F = 10 N, then determine the ratio of the tension in the rope between A and B to the tension in the rope between B and C.

__Known:__

The mass of A (m_{A}) = 1 kg

Mass B (m_{B}) = 2 kg

The mass of C (m_{C}) = 2 kg

Tensile force (F) = 10 N

__Wanted:__ T_{AB} : T_{BC}

__Solution:__

Calculate the acceleration of the system using Newton’s Second Law formula:

ΣF = m a

F = (m_{A} + m_{B} + m_{C}) a

10 = (1 + 2 + 2) a

10 = 5 a

a = 10 / 5

a = 2 m/s^{2}

Use the rope tension formula to calculate T_{AB}

ΣF = m a

T_{AB }= m_{A} a = 1 (2) = 2 Newton

Use the rope tension formula to calculate T_{BC}

ΣF = m a

T_{BC} = (m_{A} + m_{B}) a = (1 + 2) (2) = (3)(2) = 6 Newton

2. Object A with a mass of 6 kg and object B with a mass of 3 kg are connected by a rope as shown. If the coefficient of friction is 0.3 and g = 10 m/s^{2}, determine the acceleration of the object and the tension in the ropes of each block.

__Known____:__

The mass of object A (m_{A}) = 6 kg

The mass of object B (m_{B}) = 3 kg

Coefficient of friction of block A (µ_{k}) = 0.3

Acceleration due to gravity (g) = 10 m/s^{2}

The weight of block A (w_{A}) = m_{A} g = (6)(10) = 60 N

Normal force on block A (N_{A}) = w_{A} = 60 N

The weight of block B (w_{B}) = m_{B} g = (3)(10) = 30 N

__Wanted:__ The acceleration of the system (a) and the tension in the rope (T)

__Solution:__

Calculate the kinetic frictional force i.e. the frictional force when block A moves:

F_{k} = µ_{k} N_{A }= (0,3)(60) = 18 Newton

**Calculate the acceleration of the system (a):**

ΣF = m a

wB – F_{k} = (m_{A} + m_{B}) a

30 – 18 = (6 + 3) a

12 = 9 a

a = 12 / 9 = 1,3 m/s^{2}

**Calculate the tension in the string on block A (T _{A}):**

ΣF = m a

T_{A} – F_{k} = (m_{A}) a

T_{A} – 18 = (6)(1,3)

T_{A} – 18 = 7,8

T_{A} = 7,8 + 18 = 25,8 Newton

**Calculate the tension in the rope on beam B (T _{B}):**

ΣF = m a

w_{B} – T_{B }= m_{B} (a)

30 – T_{B }= 3 (1,3)

30 – T_{B }= 3,9

T_{B }= 30 – 3,9

T_{B }= 26,1 Newton

3. Two objects A and B with masses of 5 kg and 3 kg are connected by a frictionless pulley. The force P is applied to the pulley in an upward direction. If both blocks are initially at rest on the floor, what is the acceleration of block A, if the magnitude of P is 60 N?

Determine also the tension in the rope on blocks A and B.

__Known____:__

Acceleration due to gravity (g) = 10 m/s^{2}

The mass of A (m_{A}) = 5 kg

The weight of block A (w_{A}) = m_{A} g = (5)(10) = 50 Newton

Mass B (m_{B}) = 3 kg

The weight of block B (w_{B}) = m_{B} g = (3)(10) = 30 Newton

Force P = 60 N

__Wanted:__ Acceleration of the system of beams A and B (a) and the tension in the rope on beam A (T_{A}) and beam B (T_{B})

__Solution:__

**Calculate the acceleration of the system using Newton’s Second Law formula.**

ΣF = m a

w_{A} – w_{B }= (m_{A }+ m_{B}) a

50 – 30 = (5 + 3) a

20 = 8 a

a = 20 / 8

a = 2,5 m/s^{2}

**Use the tension force formula to calculate the tension in the rope**

The tension in the rope on block A:

ΣF = m a

w_{A} – T_{A} = m_{A} a

50 – T_{A }= 5 (2,5)

50 – T_{A }= 12,5

T_{A} = 50 – 12,5 = 37,5 Newton

The tension in the rope on block B:

ΣF = m a

T_{B} – w_{B} = m_{B} a

T_{B} – 30 = 3 (2,5)

T_{B} – 30 = 7,5

T_{B} = 7,5 + 30 = 37,5 Newton

20 conceptual questions and answers about rope tension.

**Question:**What is tension?**Answer:**Tension is a force that is transmitted through a string, rope, cable, or similar object when it is pulled tight by forces acting from opposite ends.**Question:**Is tension always directed along the length of the rope?**Answer:**Yes, tension always acts along the length of the rope and is directed away from the point where it is considered.**Question:**What is the tension in a rope tied to a stationary object?**Answer:**When a rope is tied to a stationary object and pulled taut, the tension in the rope is equal to the force exerted on it.**Question:**Is tension a vector quantity?**Answer:**Yes, tension is a vector quantity as it has both magnitude and direction.**Question:**Can tension in a rope ever be negative?**Answer:**No, tension cannot be negative. It’s always a pulling force, never a pushing force.**Question:**How does tension change with the angle of the rope?**Answer:**If a rope is under a load and is not perfectly vertical or horizontal, the tension in the rope must be resolved into vertical and horizontal components. The total tension remains the same, but the vertical and horizontal components vary depending on the angle.**Question:**What happens to the tension in a rope if its length is doubled?**Answer:**Doubling the length of a rope doesn’t change the tension in the rope, as long as the forces acting on it remain the same.**Question:**How is the tension in a rope affected by the mass of the object it’s lifting?**Answer:**The tension in a rope is equal to the weight of the object it is lifting if the object is in equilibrium (not accelerating). If the object’s mass increases, the tension in the rope also increases.**Question:**How does tension in a rope change when it’s under acceleration?**Answer:**When a rope is under acceleration, the tension is greater than the weight of the object. The tension in the rope is equal to the mass of the object multiplied by the acceleration, plus the weight of the object.**Question:**Can there be tension in a rope if it’s not being pulled on either end?**Answer:**No, there is no tension in a rope if it’s not being pulled or supporting a weight.**Question:**Is tension a scalar or a vector quantity?**Answer:**Tension is a vector quantity, meaning it has both magnitude (amount) and direction.**Question:**Is the tension the same at all points in a rope carrying a single stationary load?**Answer:**Yes, in an ideal case where the rope is massless and inextensible, the tension is the same at all points along the rope.**Question:**How does tension in a rope change when multiple forces act on it?**Answer:**When multiple forces act on a rope, the tension at a given point is determined by the net force acting on that section of the rope.**Question:**What factors influence the maximum tension a rope can withstand?**Answer:**The maximum tension a rope can withstand depends on its material, diameter, and construction.**Question:**How does the tension in a rope affect its length?**Answer:**When tension is applied to a rope, it will stretch and its length will increase. However, this length change is typically negligible unless the tension is close to the rope’s breaking point.**Question:**What is the relationship between tension and wave speed in a rope?**Answer:**The speed of a wave in a rope is proportional to the square root of the tension divided by the linear mass density (mass per unit length) of the rope.**Question:**What is the unit of tension?**Answer:**The SI unit of tension is the Newton (N), which is equivalent to 1 kg x m/s².**Question:**How is tension in a rope distributed in a pulley system?**Answer:**In an ideal, frictionless pulley system, the tension is the same throughout the entire length of the rope.**Question:**Can tension exist without a force being applied?**Answer:**No, tension is a reaction force that results from a force being applied to stretch or pull an object.**Question:**Does the angle of the pull affect the tension in the rope?**Answer:**The angle of the pull doesn’t affect the total tension in the rope, but it does affect the direction of the tension force. The tension force must be resolved into horizontal and vertical components if the rope is not aligned perfectly horizontally or vertically.