1. Friction question

The diagram shows a case of mass 15kg being dragged along a rough horizontal floor by means of a strap inclined at an angle A to the horizontal, where sinA = 0.6

The tension in the strap is 50N. the coefficient of friction between the case and the floor is 0.25

Modelling the case as a particle,

a) show that the normal reaction between the case and the floor has magnitude 117N,

b) find, to two decimal places, the acceleration of the case.

I have tried F=0.25*(15*0.8) + 50*0.6

this is Friction = coefficient*reaction + 50sinA
Ive no idea if this is the correct thing to do though. I get friction=66.75.

What do i do next though??

2. equilibrium in the vertical direction ...

normal force + vertical component of applied force = weight

$F_N + 50(.6) = 15g$

$F_N = 15g - 30 = 117 \, N$

net force = horizontal component of the applied force - friction force

$15a = 50(.8) - 0.25(117)$

$a = \frac{10.75}{15} = .72 \, m/s^2
$

3. Originally Posted by djmccabie
The diagram shows a case of mass 15kg being dragged along a rough horizontal floor by means of a strap inclined at an angle A to the horizontal, where sinA = 0.6

The tension in the strap is 50N. the coefficient of friction between the case and the floor is 0.25

Modelling the case as a particle,

a) show that the normal reaction between the case and the floor has magnitude 117N,

b) find, to two decimal places, the acceleration of the case.

I have tried F=0.25*(15*0.8) + 50*0.6

this is Friction = coefficient*reaction + 50sinA
Ive no idea if this is the correct thing to do though. I get friction=66.75.

What do i do next though??
Use a free body diagram and divide the forces up into horizontal and vertical components. I shall enumerate the respective components as follows:

Vertical components:
1. Normal force (upwards): $F_N = N$
2. Gravity (downwards): $F_g = mg$
3. Vertical component of the tension (upwards): $T_v = T sin \theta$

Note that the vertical components must add up to zero net force, because the block is not accelerating upwards or downwards (not levitating nor sinking into the floor). In other words, the sum of the upward forces is equal in magnitude to the sum of the downward forces. Therefore, your first equation must say: $F_g = F_N + T_v \longrightarrow mg = N + T sin \theta$

Horizontal components:
1. Horizontal component of the tension (towards rope): $T_h = T cos \theta$
2. Force of friction (against rope, as friction always opposes the direction of motion): $F_f = \mu N$

Note that the horizontal components is what gives the block its acceleration - the block is moving in a horizontal direction towards the original of the tension/pull. Therefore, the net force on the block is expressed by the difference between the horizontal tension and friction: $F_{net} = T_h - F_f = T cos \theta - \mu N$

Now, solve the two equations constructed:
#1: $mg = N + T sin \theta$
#2: $F_{net} = T cos \theta - \mu N$

From the first equation, you will find that indeed $F_N = N = 117.15$ Newtons.

To find $cos \theta$, use the identity $cos^2 \theta + sin^2 \theta = 1$, which will tell you that $cos \theta = 0.8$. Then your net force $F_{net} = T cos \theta - \mu N = 50 (0.8) - 0.25 (117.15) = 10.71$ Newtons. Using $F = ma$, we see that the acceleration is: $0.714 \frac{m}{s^2}$

In general, when solving physics problems, always try to use variables to represent your quantities. Try not to plug in numbers until the very end. This makes it easier to check over your own work if you make a mistake somewhere. It is easier to track what you're doing if you are, for example, swapping around the quantity $F_{fr} = \mu N = \mu mg$ than if you are just swapping around the number 3.14159 Newtons (then you don't know what it's supposed to represent).

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