Worksheet: Work Done by a Force Expressed in Vector Notation

In this worksheet, we will practice calculating the work done by a constant force vector acting on a body over a displacement vector using the dot product.

Q1:

A particle moves in a plane in which ⃑ 𝑖 and ⃑ 𝑗 are perpendicular unit vectors. A force, ⃑ 𝐹 = ο€Ί 9 ⃑ 𝑖 + ⃑ 𝑗  N acts on the particle. The particle moves from the origin to the point with position vector ο€Ί βˆ’ 9 ⃑ 𝑖 + 6 ⃑ 𝑗  m. Find the work done by the force.

Q2:

A force ⃑ 𝐹 = ο€Ί π‘š ⃑ 𝑖 βˆ’ 9 ⃑ 𝑗  N acts on a particle, causing a displacement ⃑ 𝑠 =  βˆ’ 5 ⃑ 𝑖 + ( π‘š + 6 ) ⃑ 𝑗  c m . If the work done by the force is 0.02 J, what is the value of π‘š ?

Q3:

A particle moved from point 𝐴 ( βˆ’ 7 , βˆ’ 1 ) to point 𝐡 ( βˆ’ 4 , 6 ) along a straight line under the action of the force ⃑ 𝐹 = π‘Ž ⃑ 𝑖 + 𝑏 ⃑ 𝑗 . During this stage of motion, the work done by the force was 106 units of work. The particle then moved from 𝐡 to another point 𝐢 ( βˆ’ 8 , βˆ’ 3 ) under the effect of the same force. During this stage of motion, the work done by the force was βˆ’ 1 3 8 units of work. Determine the two constants π‘Ž and 𝑏 .

  • A π‘Ž = βˆ’ 1 2 , 𝑏 = 1 0
  • B π‘Ž = 1 2 , 𝑏 = βˆ’ 1 0
  • C π‘Ž = βˆ’ 1 2 , 𝑏 = βˆ’ 1 0
  • D π‘Ž = 1 2 , 𝑏 = 1 0

Q4:

A particle moved from point 𝐴 ( βˆ’ 2 , βˆ’ 2 ) to point 𝐡 ( 6 , 1 0 ) along a straight line under the action of the force ⃑ 𝐹 = π‘˜ ⃑ 𝑖 βˆ’ 6 ⃑ 𝑗 acting in the opposite direction to the displacement οƒ  𝐴 𝐡 . Find the work done by the force ⃑ 𝐹 .

Q5:

A particle moved on a plane from the point 𝐴 ( βˆ’ 8 , 6 ) to the point 𝐡 ( 2 , 5 ) under the action of a force of 17 N whose line of action made an angle of πœƒ with the π‘₯ -axis where s i n πœƒ = 8 1 7 . Find the work done by this force over the displacement οƒ  𝐴 𝐡 .

  • A βˆ’ 1 4 2 or 158 work units
  • B65 or 95 work units
  • C βˆ’ 6 5 or βˆ’ 9 5 work units
  • D142 or βˆ’ 1 5 8 work units

Q6:

A particle moved from point 𝐴 ( 7 , βˆ’ 3 ) to point 𝐡 ( βˆ’ 9 , 2 ) along a straight line under the action of a force F of magnitude 8 √ 1 0 N acting in same direction as the vector c = βˆ’ 3 i βˆ’ j . Calculate the work done by the force, given that the magnitude of the displacement is measured in metres.

Q7:

A body of mass 3 kg is moving under the action of a force F , such that its displacement s ( 𝑑 ) = ο€Ή 5 𝑑   i + ( 7 𝑑 ) j . Find the work done by this force in the first 6 seconds of its motion, given that the displacement is measured in meters, the force in newtons, and the time 𝑑 in seconds.

Q8:

The displacement of a particle of mass 30 g is given as a function of time by the relation , where is a constant unit vector, is measured in centimeters, and in seconds. Given that the particle started its motion at , find the force acting on the particle and the work done by this force during the first 7 seconds of motion.

  • A dynes, ergs
  • B dynes, ergs
  • C dynes, ergs
  • D dynes, ergs

Q9:

A particle moves in a plane in which ⃑ 𝑖 and ⃑ 𝑗 are perpendicular unit vectors. Its displacement from the origin at time t seconds is given by ⃑ π‘Ÿ =  ο€Ή 2 𝑑 + 7  ⃑ 𝑖 + ( 𝑑 + 7 ) ⃑ 𝑗  2 m and it is acted on by a force ⃑ 𝐹 = ο€Ί 6 ⃑ 𝑖 + 3 ⃑ 𝑗  N . how much work does the force do between 𝑑 = 2 s and 𝑑 = 3 s ?

Q10:

The position vector of a particle of mass 3 kg moving under the action of a force is given as a function of time 𝑑 by the relation r i j ( 𝑑 ) = ( βˆ’ 4 𝑑 βˆ’ 1 0 ) + ( 3 𝑑 + 5 )   , where i and j are two perpendicular unit vectors. Calculate the work done by the force between 𝑑 = 3 to 𝑑 = 4 .

Q11:

A body of mass 2 kg is moving under the action of three forces, ⃑ 𝐹 1 , ⃑ 𝐹 2 , and ⃑ 𝐹 3 , where ⃑ 𝐹 = 𝑏 ⃑ 𝑖 βˆ’ 3 ⃑ 𝑗 1 , ⃑ 𝐹 = βˆ’ 4 ⃑ 𝑖 + 3 ⃑ 𝑗 2 , and ⃑ 𝐹 = βˆ’ 1 0 ⃑ 𝑖 + π‘Ž ⃑ 𝑗 3 , and ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors, π‘Ž and 𝑏 are constants, and each force is measured in newtons. The displacement of the body is expressed by the relation ⃑ 𝑠 ( 𝑑 ) = ο€Ή 4 𝑑  ⃑ 𝑖 + ο€Ή 3 𝑑 βˆ’ 8 𝑑  ⃑ 𝑗 2 2 , where the displacement is measured in metres, and the time 𝑑 is in seconds. Determine the work done by the resultant of the forces in the first 6 seconds of motion.

  • A βˆ’ 7 6 8 J
  • B 3 120 J
  • C 1 584 J
  • D 3 024 J

Q12:

A body moves in a plane in which ⃑ 𝑖 and ⃑ 𝑗 are perpendicular unit vectors. At time 𝑑 seconds, its position vector is given by ⃑ π‘Ÿ =  ( βˆ’ 2 𝑑 + 8 ) ⃑ 𝑖 + ο€Ή βˆ’ 𝑑 + 1 0 𝑑  ⃑ 𝑗  2 m . Given that from 𝑑 = 5 s to 𝑑 = 8 s , the change in the body’s kinetic energy was 414 J, find the body’s mass.

Q13:

A force F i j = ( βˆ’ 4 βˆ’ 9 ) N is acting on a particle whose position vector as a function of time is given by r i j ( 𝑑 ) = ο€Ή ( βˆ’ 9 𝑑 βˆ’ 8 ) + ο€Ή βˆ’ 3 𝑑 + 2    m. Calculate the work π‘Š done by the force F between 𝑑 = 3 and 𝑑 = 8 s e c o n d s .

Q14:

An object moves 10 m in the direction of . There are two forces acting on this object: N and N. Find the total work done on the object by the two forces.

Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force.

  • A Nβ‹…m
  • B Nβ‹…m
  • C Nβ‹…m
  • D Nβ‹…m
  • E 40 Nβ‹…m

Q15:

An object moves 10 m in the direction of . There are two forces acting on this object: N and N. Find the total work done on the object by the two forces.

Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force.

Q16:

An object moves 20 metres in the direction of . There are two forces acting on this object: N and N. Find the total work done on the object by the two forces.

  • A Nβ‹…m
  • B Nβ‹…m
  • C Nβ‹…m
  • D Nβ‹…m
  • E Nβ‹…m

Q17:

A body moves under the force F i j = 6 βˆ’ 9 from point 𝐴 ( βˆ’ 2 , 8 ) to point 𝐡 ( 1 , βˆ’ 7 ) . Determine the work done by the force, where the displacement is measured in metres and the force in newtons.

Q18:

A force of ⃑ 𝐹 = 6 ⃑ 𝑖 + 4 ⃑ π‘˜ n e w t o n s acts on a body and moves it from point 𝐴 ( βˆ’ 9 , βˆ’ 2 , βˆ’ 3 ) to point 𝐡 ( 8 , βˆ’ 7 , βˆ’ 1 ) . Determine the work done by the force ⃑ 𝐹 , where the displacement is measured in metres.

Q19:

A body moves from point 𝐴 ( 1 , 2 , 5 ) to 𝐡 ( 5 , 5 , βˆ’ 4 ) under a force ⃑ 𝐹 . Determine the work done if the force has magnitude √ 5 7 newtons and direction cosines ο€Ώ 5 √ 5 7 5 7 , 4 √ 5 7 5 7 , 4 √ 5 7 5 7  , taking the displacement as measured in metres.

Q20:

A particle is moving in a straight line under the action of the force ⃑ 𝐹 = βˆ’ 8 ⃑ 𝑖 βˆ’ 3 ⃑ 𝑗 from point 𝐴 ( 8 , 7 ) to point 𝐡 ( 8 , βˆ’ 5 ) . Find the work done π‘Š by the force ⃑ 𝐹 .

  • A π‘Š = 9 6 units of work
  • B π‘Š = 1 3 4 units of work
  • C π‘Š = 3 2 units of work
  • D π‘Š = 3 6 units of work

Q21:

A body is displaced by ⃑ 𝑆 = βˆ’ 6 ⃑ 𝑖 βˆ’ 5 ⃑ 𝑗 βˆ’ 7 ⃑ π‘˜ metres with a force ⃑ 𝐹 = 8 ⃑ 𝑖 βˆ’ 1 0 ⃑ 𝑗 newtons. What is the work done?

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