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Lesson: Power as the Rate of Work

Sample Question Videos

Worksheet • 16 Questions • 1 Video

Q1:

A particle is moving under the action of the force . Its position vector at time is given by the relation . Find the rate of the work done by the force at .

Q2:

A particle of unit mass is moving under the action of a force ⃑ 𝐹 = βˆ’ 2 ⃑ 𝑖 + 4 ⃑ 𝑗 . Its displacement, as a function of time, is given by ⃑ 𝑆 ( 𝑑 ) = 2 𝑑 ⃑ 𝑖 + ο€Ή 7 𝑑 + 2 𝑑  ⃑ 𝑗 2 . Find the value of 𝑑 𝑑 𝑑 ο€Ί ⃑ 𝐹 βŠ™ ⃑ 𝑆  when 𝑑 = 4 .

Q3:

A body of mass 3 kg moves under the action of a force ⃑ 𝐹 N. At time 𝑑 seconds, the velocity of the body is given by ⃑ 𝑣 =  ( βˆ’ 4 βˆ’ 2 𝑑 ) ⃑ 𝑖 + ( 5 βˆ’ 2 𝑑 ) ⃑ 𝑗  / s i n c o s m s . Find, in terms of 𝑑 , the power of the force, ⃑ 𝐹 .

  • A ( 2 4 2 𝑑 + 3 0 2 𝑑 ) c o s s i n W
  • B ( βˆ’ 6 2 𝑑 + 3 0 2 𝑑 ) c o s s i n W
  • C ( 2 4 2 𝑑 + 1 0 2 𝑑 ) c o s s i n W
  • D ( 1 2 2 𝑑 + 1 5 2 𝑑 ) c o s s i n W
  • E ( 8 2 𝑑 + 1 0 2 𝑑 ) c o s s i n W

Q4:

A body of mass 17 kg moves under the action of a force ⃑ 𝐹 . Its position vector at time 𝑑 is given by the relation ⃑ π‘Ÿ ( 𝑑 ) = ο€Ή 7 𝑑  ⃑ 𝑖 + ο€Ή 4 𝑑  ⃑ 𝑗 3 2 . Given that 𝐹 is measured in newtons, π‘Ÿ in metres, and 𝑑 in seconds, write an expression for the power of force ⃑ 𝐹 at time 𝑑 .

  • A ο€Ή 1 4 9 9 4 𝑑 + 1 0 8 8 𝑑  3 W
  • B ο€Ή 7 4 9 7 𝑑 + 5 4 4 𝑑  3 W
  • C ο€Ή 1 6 6 6 𝑑 + 2 4 4 8 𝑑  3 W
  • D ο€Ή 3 3 3 2 𝑑 + 4 8 9 6 𝑑  3 W

Q5:

A particle of mass 4 g is moving under the action of two forces ⃑ 𝐹 1 and ⃑ 𝐹 2 , where ⃑ 𝐹 = ο€Ί 6 ⃑ 𝑖 + 3 ⃑ 𝑗  1 d y n e s and ⃑ 𝐹 = ο€Ί 3 ⃑ 𝑖 + 4 ⃑ 𝑗  2 d y n e s . The position vector of the particle is given as a function of time by ⃑ π‘Ÿ ( 𝑑 ) =  ο€Ή π‘Ž 𝑑 βˆ’ 8  ⃑ 𝑖 + ο€Ή 𝑏 𝑑 + 1  ⃑ 𝑗  2 2 c m , where π‘Ž and 𝑏 are constants. Determine, in ergs per second, the power at which the resultant force acts on the particle 8 seconds after the start of motion.

Q6:

A constant force ⃑ 𝐹 , measured in dynes, is acting on a body. The displacement of the body, after 𝑑 seconds, is given by ⃑ π‘Ÿ ( 𝑑 ) = ο€Ί 2 𝑑 ⃑ 𝑖 + 3 𝑑 ⃑ 𝑗  2 c m , where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors. Given that the force ⃑ 𝐹 was working at a rate of 35 erg/s when 𝑑 = 2 , and at 43 erg/s when 𝑑 = 4 , determine ⃑ 𝐹 .

  • A ο€Ί ⃑ 𝑖 + 9 ⃑ 𝑗  dynes
  • B ο€Ί βˆ’ 9 ⃑ 𝑖 + ⃑ 𝑗  dynes
  • C ο€Ί ⃑ 𝑖 βˆ’ 9 ⃑ 𝑗  dynes
  • D ο€Ί 9 ⃑ 𝑖 + ⃑ 𝑗  dynes

Q7:

A body of mass 5 kg is moving under the action of the force F measured in newtons. Its position vector after 𝑑 seconds is given by r i j = ο€Ή 9 𝑑 + 8 𝑑  .   m Find the work done by the force F over the interval 0 ≀ 𝑑 ≀ 1 .

Q8:

A particle is moving under the action of the force . Its position vector at time is given by the relation . Find the rate of the work done by the force at .

Q9:

A particle of mass 2 g is moving under the action of two forces ⃑ 𝐹 1 and ⃑ 𝐹 2 , where ⃑ 𝐹 = ο€Ί βˆ’ 2 ⃑ 𝑖 + 3 ⃑ 𝑗  1 d y n e s and ⃑ 𝐹 = ο€Ί βˆ’ 4 ⃑ 𝑖 βˆ’ 7 ⃑ 𝑗  2 d y n e s . The position vector of the particle is given as a function of time by ⃑ π‘Ÿ ( 𝑑 ) =  ο€Ή π‘Ž 𝑑 βˆ’ 1  ⃑ 𝑖 + ο€Ή 𝑏 𝑑 βˆ’ 2  ⃑ 𝑗  2 2 c m , where π‘Ž and 𝑏 are constants. Determine, in ergs per second, the power at which the resultant force acts on the particle 6 seconds after the start of motion.

Q10:

A body of mass 2 kg moves under the action of a force ⃑ 𝐹 N. At time 𝑑 seconds, the velocity of the body is given by ⃑ 𝑣 =  ( 2 + 4 3 𝑑 ) ⃑ 𝑖 + ( βˆ’ 2 + 4 3 𝑑 ) ⃑ 𝑗  / s i n c o s m s . Find, in terms of 𝑑 , the power of the force, ⃑ 𝐹 .

  • A ( 4 8 3 𝑑 + 4 8 3 𝑑 ) c o s s i n W
  • B ( 2 4 3 𝑑 + 4 8 3 𝑑 ) c o s s i n W
  • C ( 4 8 3 𝑑 + 2 4 3 𝑑 ) c o s s i n W
  • D ( 1 6 3 𝑑 + 1 6 3 𝑑 ) c o s s i n W
  • E ( 2 4 3 𝑑 + 2 4 3 𝑑 ) c o s s i n W

Q11:

A body of mass 4 kg moves under the action of a force ⃑ 𝐹 . Its position vector at time 𝑑 is given by the relation ⃑ π‘Ÿ ( 𝑑 ) = ο€Ή 8 𝑑  ⃑ 𝑖 + ο€Ή 5 𝑑  ⃑ 𝑗 3 2 . Given that 𝐹 is measured in newtons, π‘Ÿ in metres, and 𝑑 in seconds, write an expression for the power of force ⃑ 𝐹 at time 𝑑 .

  • A ο€Ή 4 6 0 8 𝑑 + 4 0 0 𝑑  3 W
  • B ο€Ή 2 3 0 4 𝑑 + 2 0 0 𝑑  3 W
  • C ο€Ή 5 1 2 𝑑 + 9 0 0 𝑑  3 W
  • D ο€Ή 1 0 2 4 𝑑 + 1 8 0 0 𝑑  3 W

Q12:

A particle of unit mass is moving under the action of a force ⃑ 𝐹 = ⃑ 𝑖 + 5 ⃑ 𝑗 . Its displacement, as a function of time, is given by ⃑ 𝑆 ( 𝑑 ) = βˆ’ 7 𝑑 ⃑ 𝑖 + ο€Ή 5 𝑑 βˆ’ 𝑑  ⃑ 𝑗 2 . Find the value of 𝑑 𝑑 𝑑 ο€Ί ⃑ 𝐹 βŠ™ ⃑ 𝑆  when 𝑑 = 4 .

Q13:

A constant force ⃑ 𝐹 , measured in dynes, is acting on a body. The displacement of the body, after 𝑑 seconds, is given by ⃑ π‘Ÿ ( 𝑑 ) = ο€Ί 𝑑 ⃑ 𝑖 βˆ’ 5 𝑑 ⃑ 𝑗  2 c m , where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors. Given that the force ⃑ 𝐹 was working at a rate of 71 erg/s when 𝑑 = 7 , and at 31 erg/s when 𝑑 = 2 , determine ⃑ 𝐹 .

  • A ο€Ί 4 ⃑ 𝑖 βˆ’ 3 ⃑ 𝑗  dynes
  • B ο€Ί 3 ⃑ 𝑖 + 4 ⃑ 𝑗  dynes
  • C ο€Ί 4 ⃑ 𝑖 + 3 ⃑ 𝑗  dynes
  • D ο€Ί βˆ’ 3 ⃑ 𝑖 + 4 ⃑ 𝑗  dynes

Q14:

A constant force ⃑ 𝐹 , measured in dynes, is acting on a body. The displacement of the body, after 𝑑 seconds, is given by ⃑ π‘Ÿ ( 𝑑 ) = ο€Ί 2 𝑑 ⃑ 𝑖 βˆ’ 6 𝑑 ⃑ 𝑗  2 c m , where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors. Given that the force ⃑ 𝐹 was working at a rate of 6 erg/s when 𝑑 = 6 , and at 14 erg/s when 𝑑 = 7 , determine ⃑ 𝐹 .

  • A ο€Ί 2 ⃑ 𝑖 + 7 ⃑ 𝑗  dynes
  • B ο€Ί βˆ’ 7 ⃑ 𝑖 + 2 ⃑ 𝑗  dynes
  • C ο€Ί 2 ⃑ 𝑖 βˆ’ 7 ⃑ 𝑗  dynes
  • D ο€Ί 7 ⃑ 𝑖 + 2 ⃑ 𝑗  dynes

Q15:

A body of mass 3 kg is moving under the action of the force F measured in newtons. Its position vector after 𝑑 seconds is given by r i j = ο€Ή 6 𝑑 + 5 𝑑  .   m Find the work done by the force F over the interval 0 ≀ 𝑑 ≀ 1 .

Q16:

A body of mass 8 kg is moving under the action of the force F measured in newtons. Its position vector after 𝑑 seconds is given by r i j = ο€Ή 2 𝑑 + 6 𝑑  .   m Find the work done by the force F over the interval 0 ≀ 𝑑 ≀ 1 .

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