Worksheet: Newton’s Second Law: Variable Mass

In this worksheet, we will practice using differentiation for Newton’s second law of motion of a particle with variable mass.

Q1:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠=ο€Ή6𝑑+9π‘‘ο…οŠ¨m. Its mass varies with time such that π‘š=(8𝑑+9)kg. Write an expression for the force acting on the body at time 𝑑.

  • A(384𝑑+180) N
  • B(96𝑑+108) N
  • C(192𝑑+180) N
  • D(96𝑑+180) N

Q2:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠=ο€Ή2𝑑+5𝑑+4ο…οŠ¨m. Its mass varies with time such that π‘š=(6𝑑+5)kg. Determine the force acting upon the body when 𝑑=3s.

Q3:

A ball of mass 5 g was moving in a straight line through a medium loaded with dust. The dust was accumulating on its surface at a rate of 1 g/s. Find the magnitude of the force acting on the ball at time 𝑑=5seconds, given that the displacement of the ball is expressed by the relation sc(𝑑)=ο€Ό23𝑑+𝑑+7𝑑+1, where c is a unit vector in the direction of the motion and the displacement is measured in centimeters.

Q4:

The mass of a body at time 𝑑 is given by π‘š(𝑑)=(2𝑑+12)kg, whereas its position vector is rc(𝑑)=ο€Ή2𝑑+3𝑑+15ο…οŠ¨, where c is a constant unit vector, r is measured in meters, and 𝑑 is measured in seconds. Find the magnitude of the force acting on the body at 𝑑=2seconds.

Q5:

A metal ball of mass 220 g was moving in a straight line at 10 m/s through a dusty medium. If it was free from dust at the start of its motion, and the dust adhered to its surface at rate of 0.06 g/s, find the mass of the ball π‘š and the force 𝐹 acting on it at any given time 𝑑.

  • Aπ‘š=220g, 𝐹=60dynes
  • Bπ‘š=220+0.06𝑑g, 𝐹=60dynes
  • Cπ‘š=220+0.06𝑑g, 𝐹=0.6dynes
  • Dπ‘š=220𝑑+0.06g, 𝐹=0.6dynes

Q6:

A ball of mass 9 g was moving in a straight line on a dusty plane. The dust was accumulating on the ball’s surface at 2 g/s. The ball’s displacement is expressed by the relation sc(𝑑)=𝑑+3𝑑+7𝑑+2ο…οŠ©οŠ¨, where c is the unit vector in its direction of motion, 𝑑 is the time in seconds, and the magnitude of the displacement is measured in centimeters. Determine the force vector at time 𝑑, given that its magnitude is measured in dynes.

  • A(12𝑑+12)c
  • Bο€Ή18𝑑+78𝑑+72ο…οŠ¨c
  • Cο€Ή18𝑑+78𝑑+68ο…οŠ¨c
  • D(54𝑑+54)c
  • Eο€Ή6𝑑+39𝑑+68𝑑+63ο…οŠ©οŠ¨c

Q7:

A ball of mass 22 g was moving in a straight line through a medium loaded with dust. The dust was accumulating on its surface at a rate of 2 g/s. Find the magnitude of the force acting on the ball at time 𝑑=2seconds, given that the displacement of the ball is expressed by the relation sc(𝑑)=ο€Ό13𝑑+2𝑑+𝑑+8, where c is a unit vector in the direction of the motion and the displacement is measured in centimeters.

Q8:

The mass of a body at time 𝑑 is given by π‘š(𝑑)=(3𝑑+10)kg, whereas its position vector is rc(𝑑)=ο€Ή2𝑑+2𝑑+13ο…οŠ¨, where c is a constant unit vector, r is measured in meters, and 𝑑 is measured in seconds. Find the magnitude of the force acting on the body at 𝑑=2seconds.

Q9:

A metal ball of mass 104 g was moving in a straight line at 6 m/s through a dusty medium. If it was free from dust at the start of its motion, and the dust adhered to its surface at rate of 0.08 g/s, find the mass of the ball π‘š and the force 𝐹 acting on it at any given time 𝑑.

  • Aπ‘š=104g, 𝐹=48dynes
  • Bπ‘š=104+0.08𝑑g, 𝐹=48dynes
  • Cπ‘š=104+0.08𝑑g, 𝐹=0.48dynes
  • Dπ‘š=104𝑑+0.08g, 𝐹=0.48dynes

Q10:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠=ο€Ή3𝑑+8𝑑+7ο…οŠ¨m. Its mass varies with time such that π‘š=(5𝑑+4)kg. Determine the force acting upon the body when 𝑑=4s.

Q11:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠=ο€Ή9𝑑+7π‘‘ο…οŠ¨m. Its mass varies with time such that π‘š=(5𝑑+8)kg. Write an expression for the force acting on the body at time 𝑑.

  • A(360𝑑+179) N
  • B(90𝑑+144) N
  • C(180𝑑+179) N
  • D(90𝑑+179) N

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