A heavy mirror that has a width of 2 m is to be hung on a wall as shown in the figure below. The mirror weighs 700 N and the wir
e used to hang it will break if the tension exceeds 700 N (breaking strength). What is the shortest length of wire L that can be used to hang the mirror without the wire breaking? (Ignore any friction forces between the mirror and the wall.)
2 answers:
The translational equilibrium condition allows finding that the response for cable length with a maximum tension is
L = 2.56 m
Newton's second law says that the force is proportional to the mass and the acceleration of the body, in the special case that the acceleration is zero, the relationship is called the translational equilibrium condition.
∑ F = 0
Where the bold indicates vectors, F is the force and the sum is for all external forces.
The reference systems are coordinate systems with respect to which the decomposition of the vectors is carried out and the measurements are made, in this case we will use a system with the horizontal x axis and the vertical y axis.
In the attachment we can see a free body diagram of the system, let's write the equilibrium condition for each axis.
x-axis
Tₓ -Tₓ = 0
y-axis
2 - W = 0
Let's use trigonometry to decompose the tension, we can see from the graph and the adjoint that each string is half the length, let's call the angle θ
cos θ =
sin θ =
Tₓ = T cos θ
= T sin θ
We substitute
2 T sin θ = W (1)
The text indicates that the length of the block is 2 m, so the distance to the midpoint is
x = 1 m
Let's use the Pythagoras' Theorem
H² = CA² + CO²
CO =
CA = x
CO =
Where CO is the opposite leg, CA is the adjacent leg and H is the hypotenuse indicating H = L / 2,
Let's write the trigonometry functions
sin θ =
Let's substitute
sin θ =
Let's subtitute in the equation 1
2 T ( \frac{2 \sqrt{\frac{L^2}{4} -1 } }{L} ) = W
Let's solve by squaring
They indicate that the maximum tension of the cable is T = 700N and the weight is worth W = 700N, we substitute the values
L = 2.31 m
In conclusion using the translational equilibrium condition we can find that the response for cable length with maximum tension is
L = 2.31 m
Learn more about translational equilibrium here:
You might be interested in
Answer:
244.64m
Explanation:
First, we find the distance traveled with constant velocity. It's simply multiplying velocity time the time that elapsed:
After this, the ball will start traveling with a constant acceleration motion. Due to the fact that the acceleration is the opposite direction to the initial velocity, this motion will have 2 phases:
1. The velocity will start to decrease untill it reaches 0m/s.
2. Then, the velocity will start to increase at the rate of the acceleration.
The distance that the ball travels in the first phase can be found with the following expression:
Where v is the final velocity (0m/s), v_0 is the initial velocity (-8m/s) and a is the acceleration (+9m/s^2). We solve for d:
Now, before finding the distance traveled in the second phase, we need to find the time that took for the velocity to reach 0:
Then, the time of the second phase will be:
Using this, we using the equations for constant acceleration motion in order to calculate the distance traveled in the second phase:
V_0, the initial velocity of the second phase, will be 0 as previously mentioned. X_0, the initial position, will be 0, for simplicity:
So, the total distance covered by this object in meters will be the sum of all the distances we found:
Answer:
49 N
Explanation:
In order to move the box at constant speed, the acceleration of the box must be zero (a=0): this means, according to Newton's second law,
F = ma
that the net force acting on the box, F, must be zero as well.
Here there are two forces acting on the box in the horizontal direction while it is moving:
- The force of push applied by the guy, F
- The frictional force,
For an object moving on a flat surface, the frictional force is given by
where
is the coefficient of friction
m is the mass of the box
g is the acceleration of gravity
So the equation of the forces becomes
And substituting:
We find the force that must be applied by the guy: