You have a mass of 71 kg and are on a 51-degree slope hanging on to a cord with a breaking strength of 165 newtons. What must be the coefficient of static friction to 2 decimal places between you and the surface for you to be saved from the fire?
In the previous problem if the coefficient of static friction is zero, to the nearest tenth of a degree, what would the incline angle have to be in order for the cord not to break?
Well, well, well, if it isn't a fiery situation! Let's not get all hot-headed though and calculate your way out of this mess.
First, let's find out how much force is acting vertically on you due to gravity. We'll use the formula F = mg, where m is your mass (71 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F = 71 kg * 9.8 m/s^2 = 695.8 N
Now, we need to calculate the component of your weight acting down the slope. This can be found by multiplying your weight by the sine of the slope angle (51 degrees).
Component of weight down the slope = 695.8 N * sin(51) = 530.9 N
To prevent you from sliding down, the force of static friction must be equal to this component of weight down the slope. So, we have:
Force of static friction = 530.9 N
The maximum possible force of static friction is given by the coefficient of static friction (let's call it µ) multiplied by the normal force acting against you. The normal force is equal to your weight (695.8 N) multiplied by the cosine of the slope angle (51 degrees).
Maximum force of static friction = µ * 695.8 N * cos(51)
For the cord to save you from the fire, the maximum force of static friction should be equal to or greater than the force of static friction required to counteract your weight down the slope.
So, we have the following inequality:
µ * 695.8 N * cos(51) ≥ 530.9 N
Solving this inequality for µ, we find:
µ ≥ 530.9 N / (695.8 N * cos(51))
µ ≥ 0.86 (approximately)
Therefore, the coefficient of static friction must be at least 0.86 for you to be saved from the fire. Hang in there, my friend! 🤡
Now, onto the second part of your question. If the coefficient of static friction is zero, it means that there is no friction between you and the surface. In this case, your only concern is the breaking strength of the cord.
To find the incline angle at which the cord wouldn't break, we need to calculate the component of your weight perpendicular to the slope. This can be found using the formula:
Component of weight perpendicular to the slope = mass * gravity * cos(angle)
Since we don't want the cord to break, this component of weight should be less than or equal to the breaking strength of the cord.
165 N ≥ 71 kg * 9.8 m/s^2 * cos(angle)
Let's solve this equation for angle:
cos(angle) ≤ 165 N / (71 kg * 9.8 m/s^2)
angle ≤ arccos(165 N / (71 kg * 9.8 m/s^2))
Calculating this, we find:
angle ≤ 68.5 degrees (approximately)
Therefore, the incline angle would have to be less than or equal to 68.5 degrees (to the nearest tenth of a degree) in order for the cord not to break. Keep your angle of incline low and your hopes high! 🤡
To calculate the coefficient of static friction, we need to consider the forces acting on you while hanging on the cord. The forces in this situation are your weight (mg) and the tension in the cord (T).
1. Determine the force of gravity acting on you:
Fg = mg
Fg = 71 kg * 9.8 m/s^2
Fg = 696.8 N
2. Resolve the force of gravity along the slope:
Fg_parallel = Fg * sin(angle)
Fg_parallel = 696.8 N * sin(51 degrees)
Fg_parallel = 535.3 N
3. Calculate the normal force (Fn):
Fn = Fg_parallel
Fn = 535.3 N
4. Determine the maximum static frictional force (Fs_max):
Fs_max = coefficient of static friction * Fn
Let's denote the coefficient of static friction as μ.
Fs_max = μ * Fn
5. Compare the maximum static frictional force to the breaking strength of the cord:
Fs_max ≤ breaking strength
Therefore:
μ * Fn ≤ 165 N
Plugging in the values, we can solve for the coefficient of static friction (μ).
μ * 535.3 N ≤ 165 N
μ ≤ 165 N / 535.3 N
μ ≤ 0.308 to 3 decimal places
Hence, the coefficient of static friction must be less than or equal to 0.308 for you to be saved from the fire.
For the second question, if the coefficient of static friction is zero, the tension in the cord will solely depend on the angle of the slope and the weight of the person.
Let's denote the maximum tension in the cord as T_max and the incline angle as α.
1. Calculate the force of gravity acting on you:
Fg = mg
Fg = 71 kg * 9.8 m/s^2
Fg = 696.8 N
2. Resolve the force of gravity perpendicular to the slope:
Fg_perpendicular = Fg * cos(α)
Fg_perpendicular = 696.8 N * cos(α)
3. As there is no friction, the tension in the cord must equal the perpendicular component of gravity:
T_max = Fg_perpendicular
T_max = 696.8 N * cos(α)
To find the angle at which the cord will not break, we need to solve the equation:
T_max =breaking strength
696.8 N * cos(α) = 165 N
Solving for α:
cos(α) = 165 N / 696.8 N
α = cos^(-1)(165 N / 696.8 N)
Calculating this value:
α = 77.4 degrees (to the nearest tenth of a degree)
Therefore, if the coefficient of static friction is zero, the incline angle needs to be at least 77.4 degrees for the cord not to break.
To find the coefficient of static friction between you and the surface, we'll first determine the maximum force of static friction that can be applied before the cord breaks.
The force of gravity acting on you can be calculated using the formula:
Force of gravity = mass × acceleration due to gravity
= 71 kg × 9.8 m/s^2
= 695.8 N
Next, we need to calculate the component of the force of gravity parallel to the slope, which can be found by multiplying the force of gravity by the sine of the slope angle:
Force parallel to slope = Force of gravity × sin(slope angle)
= 695.8 N × sin(51°)
≈ 528.2 N
Since the cord can withstand a maximum force of 165 N, the maximum force of static friction that can be applied is 165 N.
The force of static friction can be calculated using the formula:
Force of static friction = coefficient of static friction × normal force
Since the normal force is equal to the force perpendicular to the slope, we can calculate it using the formula:
Normal force = Force of gravity × cos(slope angle)
= 695.8 N × cos(51°)
≈ 447.4 N
Now, we can calculate the coefficient of static friction:
Coefficient of static friction = Force of static friction / Normal force
= 165 N / 447.4 N
≈ 0.37 (to two decimal places)
Therefore, the coefficient of static friction between you and the surface needs to be approximately 0.37 for you to be saved from the fire.
In the scenario where the coefficient of static friction is zero, the maximum force of static friction that can be applied is zero. Therefore, the force parallel to the slope should be equal to or less than the breaking strength of the cord.
Using the same formula as before:
Force parallel to slope = Force of gravity × sin(slope angle)
= 695.8 N × sin(angle)
To find the angle that satisfies the condition, we set the force parallel to slope equal to the breaking strength of the cord:
695.8 N × sin(angle) = 165 N
Solving for the angle:
sin(angle) = 165 N / 695.8 N
angle ≈ arcsin(0.237) ≈ 13.6 degrees (to the nearest tenth of a degree)
Therefore, the incline angle would have to be approximately 13.6 degrees (to the nearest tenth of a degree) for the cord not to break if the coefficient of static friction is zero.
x: 0= m•g•sinα-T-F(fr),
y: 0 = - m•g•cos α +N.
F(fr) =k•N=k• m•g•cos α,
F(fr) = m•g•sinα – T,
k• m•g•cos α = m•g•sinα – T,
k= tan α – T/m•g•cos α =
=tan51º- 165/71•9.8•cos51 =
= 0.86.
If k=0,
tan α = T/m•g•cos α,
sin α/ cos α = T/m•g•cos α,
sin α = T/m•g =167/71•9.8 =0.24
α =13.88º