Lesson Worksheet: Power as the Rate of Work Mathematics

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In this worksheet, we will practice defining the power of a force as the derivative of the work done by the force.

Q1:

A particle is moving under the action of the force Fij=8+5. Its position vector at time 𝑡 is given by the relation rij(𝑡)=3𝑡−𝑡−4+52𝑡+10𝑡+5. Find the rate of the work done by the force at 𝑡=3.

Q2:

A particle of unit mass is moving under the action of a force Fij=−2+4. Its displacement, as a function of time, is given by Sij(𝑡)=2𝑡+7𝑡+2𝑡. Find the value of dd𝑡(⋅)FS when 𝑡=4.

  • A60 power units
  • B228 power units
  • C476 power units
  • D124 power units

Q3:

A body of mass 3 kg moves under the action of a force F N. At time 𝑡 seconds, the velocity of the body is given by vij=[(−4−2𝑡)+(5−2𝑡)]/sincosms. Find, in terms of 𝑡, the power of the force, F.

  • A(−62𝑡+302𝑡)cossin W
  • B(242𝑡+102𝑡)cossin W
  • C(82𝑡+102𝑡)cossin W
  • D(122𝑡+152𝑡)cossin W
  • E(242𝑡+302𝑡)cossin W

Q4:

A body of mass 17 kg moves under the action of a force F. Its position vector at time 𝑡 is given by the relation rij(𝑡)=7𝑡+4𝑡. Given that 𝐹 is measured in newtons, 𝑟 in meters, and 𝑡 in seconds, write an expression for the power of force F at time 𝑡.

  • A3,332𝑡+4,896𝑡 W
  • B7,497𝑡+544𝑡 W
  • C14,994𝑡+1,088𝑡 W
  • D1,666𝑡+2,448𝑡 W

Q5:

A particle of mass 4 g is moving under the action of two forces F and F, where Fij=(6+3)dynes and Fij=(3+4)dynes. The position vector of the particle is given as a function of time by rij(𝑡)=𝑎𝑡−8+𝑏𝑡+1cm, where 𝑎 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 𝑡 ‏seconds, is given by rij(𝑡)=2𝑡+3𝑡cm, where i and j are two perpendicular unit vectors. Given that the force F was working at a rate of 35 erg/s when 𝑡=2, and at 43 erg/s when 𝑡=4, determine F.

  • A(9+)ijdynes
  • B(−9+)ijdynes
  • C(−9)ijdynes
  • D(+9)ijdynes

Q7:

A body of mass 5 kg is moving under the action of the force F measured in newtons. Its position vector after 𝑡 seconds is given by rij=9𝑡+8𝑡.m Find the work done by the force F over the interval 0≤𝑡≤1.

Q8:

The power of a machine is given by the relation 𝑃=3𝑡+7, where 𝑡 is the time elapsed in seconds. Find the work done by the machine in the first 8 seconds.

Q9:

The power of an engine is given by 8𝑡−115𝑡 hp, where 𝑡∈[0,103] is the time in seconds. Find the power of the engine 𝑃 when 𝑡=36seconds, the work done 𝑊 over the time interval [0,12], and the maximum power 𝑃max of the engine.

  • A𝑃=201.6hp, 𝑊=537.6⋅kg-wtm, 𝑃=1,200maxhp
  • B𝑃=201.6hp, 𝑊=40,320⋅kg-wtm, 𝑃=240maxhp
  • C𝑃=489.6hp, 𝑊=537.6⋅kg-wtm, 𝑃=720maxhp
  • D𝑃=547.2hp, 𝑊=77,760⋅kg-wtm, 𝑃=840maxhp

Q10:

The power of an engine at time 𝑡 seconds is given by 𝑃(𝑡)=12𝑡+8𝑡W. Find the work done by the engine between 𝑡=2s and 𝑡=3s.

This lesson includes 17 additional questions and 185 additional question variations for subscribers.

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