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In this lesson, we will learn how to solve problems using Newton's second law for a body of variable motion and/or one that is acted upon by a variable force.

Q1:

A body moves in a straight line. At time π‘ seconds, its displacement from a fixed point is given by π = οΉ 6 π‘ + 9 π‘ ο ο¨ m . Its mass varies with time such that π = ( 8 π‘ + 9 ) k g . Write an expression for the force acting on the body at time π‘ .

Q2:

A body moves in a straight line. At time π‘ seconds, its displacement from a fixed point is given by π = οΉ 2 π‘ + 5 π‘ + 4 ο ο¨ m . Its mass varies with time such that π = ( 6 π‘ + 5 ) k g . Determine the force acting upon the body when π‘ = 3 s .

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 π‘ = 5 s e c o n d s , given that the displacement of the ball is expressed by the relation β π ( π‘ ) = οΌ 2 3 π‘ + π‘ + 7 π‘ + 1 ο β π 3 2 , where β π is a unit vector in the direction of the motion and the displacement is measured in centimetres.

Q4:

A body of mass 6 kg is initially at rest at a point π . It starts to move under the action of a force πΉ ( π₯ ) = ( 2 π₯ + 8 ) N where π₯ m is the displacement of the body from π . Find the displacement of the body when its velocity is π£ = 4 / m s .

Q5:

A body of mass 2 kg moves along a straight line under the action of a force, πΉ . The force acting on the body is πΉ = ( 9 π₯ + 7 ) N, where π₯ is the displacement of the body from its initial position. Determine velocity of the body when π₯ = 4 m .

Q6:

A body moves in a straight line under the action of a force, πΉ = οΌ 2 5 π£ + 3 ο N , where π£ is the velocity of the body at time π‘ seconds. If the initial velocity of the body is 3 m/s, at what time does the body reach a speed of 7 m/s?

Q7:

The mass of a body at time π‘ is given by π ( π‘ ) = ( 2 π‘ + 1 2 ) k g , whereas its position vector is β π ( π‘ ) = οΉ 2 π‘ + 3 π‘ + 1 5 ο β π 2 , where β π is a constant unit vector, β π is measured in metres, and π‘ is measured in seconds. Find the magnitude of the force acting on the body at π‘ = 2 seconds.

Q8:

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 and the force πΉ acting on it at any given time π‘ .

Q9:

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

Q10:

A body of mass 20 kg started moving from rest along the π₯ -axis. When the bodyβs displacement relative to the origin was π m in the direction of π₯ increasing, it moved under the effect of a force given by πΉ = ( 1 0 π + 5 ) N . Find the bodyβs velocity at π = 8 m .

Q11:

A body of mass 11 kg started moving along the π₯ -axis at an initial velocity 8 m/s in the direction of π₯ increasing. After time π‘ seconds, where π‘ β₯ 0 , the bodyβs velocity was π£ m/s in the same direction, and the force acting on the body was πΉ = οΌ 4 6 π£ + 6 ο N . Determine the value of π‘ at which π£ = 1 0 / m s .

Q12:

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 π‘ = 2 s e c o n d s , given that the displacement of the ball is expressed by the relation β π ( π‘ ) = οΌ 1 3 π‘ + 2 π‘ + π‘ + 8 ο β π 3 2 , where β π is a unit vector in the direction of the motion and the displacement is measured in centimetres.

Q13:

The mass of a body at time π‘ is given by π ( π‘ ) = ( 3 π‘ + 1 0 ) k g , whereas its position vector is β π ( π‘ ) = οΉ 2 π‘ + 2 π‘ + 1 3 ο β π 2 , where β π is a constant unit vector, β π is measured in metres, and π‘ is measured in seconds. Find the magnitude of the force acting on the body at π‘ = 2 seconds.

Q14:

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 and the force πΉ acting on it at any given time π‘ .

Q15:

A body moves in a straight line. At time π‘ seconds, its displacement from a fixed point is given by π = οΉ 3 π‘ + 8 π‘ + 7 ο ο¨ m . Its mass varies with time such that π = ( 5 π‘ + 4 ) k g . Determine the force acting upon the body when π‘ = 4 s .

Q16:

A body moves in a straight line. At time π‘ seconds, its displacement from a fixed point is given by π = οΉ 6 π‘ + 2 π‘ ο ο¨ m . Its mass varies with time such that π = ( 5 π‘ + 3 ) k g . Write an expression for the force acting on the body at time π‘ .

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