The system shown in the figure is in equilibrium. A 15 kg mass is on the table. A string
attached to the knot and the ceiling makes an
angle of 54◦
with the horizontal. The coefficient of the static friction between the 15 kg
mass and the surface on which it rests is 0.4
What is the largest mass m can have and
still preserve the equilibrium? The acceleration of gravity is 9.8 m/s^2
Answer in units of kg
Help is appreciated guys
To find the largest mass m that can still preserve equilibrium, we can start by analyzing the forces acting on the system.
1. Weight: The 15 kg mass exerts a downward force due to gravity, given by the equation Fg = m × g, where m is the mass and g is the acceleration due to gravity (9.8 m/s^2). Therefore, the weight of the 15 kg mass is Fg = 15 kg × 9.8 m/s^2.
2. Tension: The tension in the string pulls the attached mass in the opposite direction to counterbalance the weight. The tension in the string is the same at both ends, so it cancels out.
3. Friction: The static friction force between the 15 kg mass and the surface opposes any tendency for the mass to slide. The maximum static friction force is given by the equation Ff = μ × N, where μ is the coefficient of static friction (0.4) and N is the normal force. The normal force is equal to the weight of the 15 kg mass because the system is in equilibrium and not accelerating vertically.
Now, let's set up the equilibrium equations:
In the horizontal direction:
Tension - Friction = 0
In the vertical direction:
Weight - Tension = 0
Since the angle between the string and the horizontal is 54 degrees, we need to take components of the weight and tension forces.
The vertical component of tension is given by T_V = T × sin(54°), and the vertical component of the weight is given by W_V = 15 kg × g × cos(54°).
Setting up the equations:
T × sin(54°) - μ × N = 0
15 kg × g × cos(54°) - T = 0
Substituting N with mg (since N is equal to the weight of the 15 kg mass):
T × sin(54°) - μ × mg = 0
15 kg × g × cos(54°) - T = 0
Now we can solve these equations simultaneously to find the value of T.
Knowing T, we can find the maximum value of m that still preserves equilibrium by setting T = μ × mg and solving for m.
Let's calculate the value of T:
T × sin(54°) - 0.4 × 15 kg × 9.8 m/s^2 = 0
T = 0.4 × 15 kg × 9.8 m/s^2 ÷ sin(54°)
Now we can find the maximum mass m:
T ≈ 98.733 N
m = T ÷ (μ × g)
m ≈ 98.733 N ÷ (0.4 × 9.8 m/s^2)
Calculating this, we find:
m ≈ 25.34 kg
Therefore, the largest mass m that can still preserve equilibrium is approximately 25.34 kg.