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, Fij=(9+)N acts on the particle. The particle moves from the origin to the point with position vector (βˆ’9+6)ij m. Find the work done by the force.

Q2:

A force Fij=(π‘šβˆ’9)N acts on a particle, causing a displacement sij=[βˆ’5+(π‘š+6)]cm. 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 Fij=π‘Ž+𝑏. 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’138 units of work. Determine the two constants π‘Ž and 𝑏.

  • Aπ‘Ž=12, 𝑏=10
  • Bπ‘Ž=12, 𝑏=βˆ’10
  • Cπ‘Ž=βˆ’12, 𝑏=10
  • Dπ‘Ž=βˆ’12, 𝑏=βˆ’10

Q4:

A particle moved from point 𝐴(βˆ’2,βˆ’2) to point 𝐡(6,10) along a straight line under the action of the force Fij=π‘˜βˆ’6 acting in the opposite direction to the displacement 𝐴𝐡. 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 sinπœƒ=817. Find the work done by this force over the displacement 𝐴𝐡.

  • A142 or βˆ’158 work units
  • Bβˆ’65 or βˆ’95 work units
  • Cβˆ’142 or 158 work units
  • D65 or 95 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√10 N acting in the same direction as the vector cij=βˆ’3βˆ’. Calculate the work done by the force, given that the magnitude of the displacement is measured in meters.

Q7:

A body of mass 3 kg is moving under the action of a force F, such that its displacement sij(𝑑)=ο€Ή5𝑑+(7𝑑). 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 si(𝑑)=ο€Ή5π‘‘βˆ’6π‘‘ο…οŠ¨, where i is a constant unit vector, 𝑠 is measured in centimeters, and 𝑑 in seconds. Given that the particle started its motion at 𝑑=0, find the force 𝐹 acting on the particle and the work done π‘Š by this force during the first 7 seconds of motion.

  • A𝐹=βˆ’210idyn, π‘Š=54,390erg
  • B𝐹=βˆ’355idyn, π‘Š=91,945erg
  • C𝐹=βˆ’30idyn, π‘Š=7,770erg
  • D𝐹=βˆ’360idyn, π‘Š=93,240erg

Q9:

A particle moves in a plane in which i and j are perpendicular unit vectors. Its displacement from the origin at time 𝑑 seconds is given by rij=2𝑑+7+(𝑑+7)m and it is acted on by a force Fij=(6+3)N. How much work does the force do between 𝑑=2s and 𝑑=3s?

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 rij(𝑑)=(βˆ’4π‘‘βˆ’10)+(3𝑑+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, F, and F, where Fij=π‘βˆ’3, Fij=βˆ’4+3, and Fij=βˆ’10+π‘Ž, 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 sij(𝑑)=ο€Ή4𝑑+ο€Ή3π‘‘βˆ’8π‘‘ο…οŠ¨οŠ¨, where the displacement is measured in meters, and the time 𝑑 is in seconds. Determine the work done by the resultant of the forces in the first 6 seconds of motion.

Q12:

A force Fij=(βˆ’4βˆ’9) N is acting on a particle whose position vector as a function of time is given by rij(𝑑)=ο€Ή(βˆ’9π‘‘βˆ’8)+ο€Ήβˆ’3𝑑+2ο…ο…οŠ¨ m. Calculate the work π‘Š done by the force F between 𝑑=3 and 𝑑=8seconds.

Q13:

An object moves 10 m in the direction of ji+. There are two forces acting on this object: Fijk=+2+2 N and Fijk=5+2βˆ’6 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.

  • A30√2 Nβ‹…m
  • B10√2 Nβ‹…m
  • C50√2 Nβ‹…m
  • D40 Nβ‹…m
  • E60√2 Nβ‹…m

Q14:

An object moves 10 m in the direction of j. There are two forces acting on this object: Fijk=++2 N and Fijk=βˆ’5+2βˆ’6 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.

Q15:

An object moves 20 meters in the direction of kj+. There are two forces acting on this object: Fijk=++2 N and Fijk=+2βˆ’6 N. Find the total work done on the object by the two forces.

  • Aβˆ’20√2 Nβ‹…m
  • B20√2 Nβ‹…m
  • C50√2 Nβ‹…m
  • Dβˆ’10√2 Nβ‹…m
  • E10√2 Nβ‹…m

Q16:

A body moves under the force Fij=6βˆ’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 meters and the force in newtons.

Q17:

A force of Fik=6+4newtons 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.

Q18:

A body moves from point 𝐴(1,2,5) to point 𝐡(5,5,βˆ’4) under a force F. Determine the work done if the force has a magnitude of √57 newtons and direction cosines ο“’5√5757,4√5757,4√5757ο““, taking the displacement as measured in meters.

Q19:

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

  • Aπ‘Š=32 units of work
  • Bπ‘Š=36 units of work
  • Cπ‘Š=96 units of work
  • Dπ‘Š=96 units of work
  • Eπ‘Š=134 units of work

Q20:

A body is displaced by Sijk=βˆ’9+2+10 m with a force Fijk=βˆ’9+2βˆ’3 N. What is the work done?

Q21:

A force of 8 newtons moves a body from 𝐴(2,2,6) to 𝐡(3,βˆ’7,5). Taking 1 unit to equal 1 meter, what is the work done?

  • A8√83 J
  • B24 J
  • C664 J
  • Dβˆ’72 J

Q22:

A body moves in a plane in which i and j are perpendicular unit vectors. Two forces, Fij=(9βˆ’2)N and Fij=(9βˆ’7)N act on the body. The particle moves from the point with position vector (βˆ’6+2)ij m to the point (2+3)ij m. Find the work done by the resultant of the forces.

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