A 180-kg crate is sitting on the flatbed of a moving truck. The coefficient of sliding friction between the crate and the truck bed is 0.30. Two taut cables are attached to either side of the crate. Each cable can exert a maximum horizontal force of 650 N either forward or backward if the crate begins to slide. If the truck stops in 1.8 s, what is the maximum speed the truck could have been moving without breaking the cables?
18 m/s
r3
To determine the maximum speed the truck could have been moving without breaking the cables, we can use the concept of static friction.
Step 1: Understand the problem.
- A 180-kg crate is sitting on the flatbed of a moving truck.
- The coefficient of sliding friction between the crate and the truck bed is 0.30.
- Two taut cables are attached to either side of the crate.
- Each cable can exert a maximum horizontal force of 650 N either forward or backward if the crate begins to slide.
- The truck stops in 1.8 s.
Step 2: Determine the normal force.
The normal force acting on the crate is equal in magnitude but opposite in direction to the weight of the crate. The weight can be calculated by multiplying the mass of the crate by the acceleration due to gravity (9.8 m/s²).
Weight = 180 kg × 9.8 m/s² = 1764 N
Therefore, the normal force acting on the crate is 1764 N.
Step 3: Calculate the maximum force of static friction.
The maximum force of static friction can be calculated by multiplying the coefficient of sliding friction by the normal force.
Maximum force of static friction = (coefficient of sliding friction) × (normal force)
= 0.30 × 1764 N
= 529.2 N
Step 4: Calculate the acceleration of the truck.
The acceleration of the truck can be calculated using Newton's second law of motion, F = ma, where F is the net force acting on the crate, m is the mass of the crate, and a is the acceleration.
Since the crate is coming to a stop, the net force is equal to the force of static friction.
Force of static friction = net force = ma
529.2 N = 180 kg × a
Solving for a gives:
a = 529.2 N / 180 kg
a = 2.94 m/s²
Step 5: Calculate the maximum speed of the truck.
The maximum speed of the truck can be calculated by using the equation for uniformly accelerated motion, v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken.
Here, the truck stops, so the final velocity is 0.
0 = u + (2.94 m/s²)(1.8 s)
Solving for u gives:
u = -5.29 m/s
Since we are interested in the maximum speed the truck could have been moving without breaking the cables, we take the absolute value:
Maximum speed = |u| = 5.29 m/s
Therefore, the maximum speed the truck could have been moving without breaking the cables is 5.29 m/s.
To determine the maximum speed the truck could have been moving without breaking the cables, we need to consider the forces acting on the crate and use the concept of Newton's second law of motion.
First, let's identify the forces acting on the crate:
1. Weight (mg): This is the force of gravity pulling the crate downward, given by the equation Fg = mg, where m is the mass of the crate (180 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2). So, Fg = 180 kg * 9.8 m/s^2 = 1764 N.
2. Normal force (N): This is the force exerted by the truck bed on the crate in the vertical direction, which balances the weight of the crate. Since the crate is on a flatbed, the normal force N is equal to the weight of the crate, so N = 1764 N.
Now, let's analyze the horizontal forces acting on the crate when it starts to slide:
3. Sliding friction force (fs): The force of sliding friction opposes the motion of the crate. It is given by the equation fs = μ * N, where μ is the coefficient of sliding friction (0.30) and N is the normal force. So, fs = 0.30 * 1764 N = 529.2 N.
We want to determine the maximum velocity (speed) the truck could have before the force applied by the cables is exceeded. To calculate this, we'll apply Newton's second law of motion, which states that the net force on an object is equal to the mass of the object times its acceleration (Fnet = ma).
In this case, the net force is given by the difference between the force applied by the cables and the force of sliding friction:
Fnet = Fcable - fs,
where Fcable is the force applied by the cables and fs is the force of sliding friction.
Since we want to find the maximum speed, we can assume the acceleration is constant throughout the 1.8-second time interval. Therefore, we can rewrite Newton's second law as:
Fnet = m * (final velocity - initial velocity) / time,
where the final velocity is 0 m/s (since the truck stops) and the initial velocity is what we're trying to find. We can rearrange the equation to solve for the initial velocity:
initial velocity = (Fnet * time) / m.
Substituting the values, we have:
initial velocity = ((650 N - 529.2 N) * 1.8 s) / 180 kg.
Calculating this, we get:
initial velocity = (120.8 N * 1.8 s) / 180 kg
= 12.08 m/s.
Therefore, the maximum speed the truck could have been moving without breaking the cables is 12.08 m/s.