Please show every step of the solutions. Thanks in advance! Problem 1. A passeng

Question

Please show every step of the solutions. Thanks in advance!

Problem 1. A passenger sitting in the rear of a bus claims that he was injured when the driver slammed on the brakes, causing a suitcase to come flying toward him from the front of the bus. If you were the judge in this case, what disposition would you make? Explain your reasoning. Problem 2. Four forces have the same magnitudes but different directions as shown in the figures to the right. A mass initially at rest is suddenly acted upon by forces 1 and 2. A second identical mass initially at rest is suddenly acted upon by forces 3 and 4. Compare the magnitudes and directions of their accelerations. Explain. Problem 3. Suppose a truck loaded with sand accelerates along a highway. If the driving force exerted on the truck remains constant, what happens to the truck's acceleration if its trailer leaks sand at a constant rate? Problem 4. A 10 kg block rests on the floor. (a) What force does the floor exert on the block? (b) A rope is tied to the block and is run vertically over a pulley. The other end of the rope is attached to a free hanging 5 kg object. What is the force exerted by the floor on the 10 kg block? (c) The 5 kg object is replaced by a 15 kg object. What is the force exerted by the floor now? Problem 5. A 3 kg object is moving in a plane, with its x and y coordinates given by x = 5t^2 - 1 and y = 3t^3 + 2, where x and y are in meters and t is in seconds. Find the magnitude of the net force acting on this object as a function of time. Evaluate this expression at t = 2 s.

Explanation / Answer

1) considering the newton's law of inertia

now imagine the passanger in travelling in the bus which has a constant velocity.

so the velocity of the passenger and the suitcase inside the bus is the same with respect to ground.

now the moment the driver hits the brake.the velocity of the bus becomes zero making the speed of the passenger and the suitcase to zero with respect to ground.

so they both refuse to undergo the change and according to newton's law of inertia they tend to move forward even though the brakes are applied.

if the suitcase hits then the suitcase it would be the one behind him, but not the one infront of him,

so he cannot see the suitcase flying infront of him and hit him,

so that is a false accusation.

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2) consider the case one where F1 and F2 are acting on the body.

each force can be divided into x component and y component.

x xomponent of F1 will be F1cos(theta) along -x axis and y component will be F1sin(theta) along +y axis.

similalarly F2 has F2cos(theta) along +x axis and F2sin(theta) along + y axis.

since F1 = F2 therefore the x components cancel out each other leaving the y components.

therefore net force will be (F1 + F2 )sin(theta)

and since F1 = F2 = F3 + F4 = F(say)

we get net force = 2Fsin(theta)

acceleration will be = 2Fsin(theta)/m  if m is the mass.

similarly for case 2 where F3 and F4 are acting we get

net force = 2Fsin(phi) and acceleration will be 2Fsin(phi)/m

now the ratio of there acceleration will be sin(theta)/sin(phi).

and both accelerations will be in the same direction ie., + y axis

so ratio of directions will be 1

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3) we know that acceleration = Force / m.

here we are given that force which is driving it is kept constant

but the sand is leaking at a constant rate.

which thereby decreases the mass at a constant rate.

which therefore increases the truck's acceleration at a constant rate

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4) a) force exerted by the floor on the block is the same as the force exerted by the block on the floor but in opposite direction.

force exerted by the block is mg downwards.

so force exerted by the floor on block is 10g= 98.1 N but upwards.

b) now here the pulley mechanism is trying to lift the block, but the force exerted by the pulley mechanism is only 5g upwards.

net forces acting on the block are 5g upwards and the force exerted by the floor upwards(F floor)

net forces acted by the block are 10g downwards.

equating these 2 we get 5g + F(floor) = 10g.

this impies F floor = 5g = 5*9.81 = 49.05 N upwards

force exerted by the floor on block is 49.05 N and it is upwards.

c) now here the force exerted by the pulley mechanism is 10g.

therefrore equating again we get 10g + F ( floor) = 10g.

therefore F(floor) = 0.

so force exerted by the floor on block is 0 N

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5) we know that force = mass * acceleration.

we know mass = 3 kg,

now we have to find acceleration in both directions.

given the equation for displacement.

we can get the equation for acceeleration by double derivating it.

x = 5t^2 - 1 , by derivating once we get x' = 10t and by double derivating x" = 10.

similarly y = 3t^3 + 2 , y' = 9t^2 and y" = 18t.

force along x axis is mass * acceleration = 3*10 = 30 N.

that along y axis is = 3* 18t

now net magnitude is sqrt(30^2 + (54t)^2)

now if we put t = 2

we get net magnitude is sqrt(30^2 + (54*2)^2)

therefore net magnitude is 112.089 N at t=2 seconds

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