A pickup truck is carrying a toolbox, but the rear gate of the truck is missing, so the box will slide out if it is set moving. The coefficients of kinetic and static friction between the box and the bed of the truck are 0.340 and 0.750, respectively.
Starting from rest, what is the shortest time this truck could accelerate uniformly to 33.0 m/s (≈ 73.8 mph ) without causing the box to slide. (Hint: First use Newton’s second law to find the maximum acceleration that static friction can give the box, and then solve for the time required to reach 33.0 m/s .)
normal force = m g
max static = .75 m g
so m a = .75 m g for slide to start
or a = .75 g
now the slide
a = .75 g
v = a t
33 = .75 g t
t = 33/(.75*9.81)
What happens to the m?
Well, well, well, looks like this truck has some sliding issues! Let's see what we can do to help.
To start off, we need to find the maximum acceleration that static friction can provide. Newton's second law tells us that the force of static friction is equal to the mass of the box multiplied by the acceleration. The maximum force of static friction can be found by multiplying the coefficient of static friction (0.750) by the normal force, which is equal to the weight of the box.
Now, let's shift gears and calculate the maximum force of static friction. But wait, what's that? I see your puzzled expression. Don't worry, I've got this covered! The normal force is equal to the weight of the box, which can be found by multiplying the mass of the box by the acceleration due to gravity (9.8 m/s²).
Once we've found the maximum force of static friction, we can divide it by the mass of the box to get the maximum acceleration it can provide. And voila! We've got our maximum acceleration.
Next on the agenda is finding the time it takes for the truck to reach a velocity of 33.0 m/s. We can use the equation of motion: final velocity equals initial velocity plus acceleration multiplied by time. Since the truck starts from rest (initial velocity equals zero), we can rearrange the equation to solve for time.
Now, plug in the maximum acceleration we found earlier and the final velocity of 33.0 m/s. Crunch those numbers, and there you have it – the shortest time it would take for this truck to reach 33.0 m/s without having the box slide.
But remember, be sure to check if the acceleration we found is less than the maximum acceleration of kinetic friction. If it's not, well, you might have a flying toolbox on your hands. And that's definitely not funny! Keep it safe, my friend!
To find the shortest time the truck can accelerate uniformly to 33.0 m/s without causing the box to slide, we can use Newton's second law and the concept of static friction.
1. First, let's find the maximum acceleration the static friction can provide. The formula for static friction is given by:
fs = μs * N
where fs is the force of static friction, μs is the coefficient of static friction, and N is the normal force.
2. The normal force is equal to the weight of the toolbox, which can be calculated as:
N = m * g
where m is the mass of the toolbox and g is the acceleration due to gravity (approximately 9.8 m/s^2).
3. Now, substituting the values for the coefficient of static friction and the normal force into the formula for static friction, we get:
fs = μs * N
= μs * (m * g)
This is the maximum force of static friction that can be applied to the box.
4. According to Newton's second law, the net force applied on the box is equal to the mass of the box times its acceleration:
Fnet = m * a
where Fnet is the net force and a is the acceleration.
5. In this case, the net force is the force provided by static friction: Fnet = fs = μs * (m * g)
Thus, we have:
μs * (m * g) = m * a
Simplifying the equation, we get:
a = μs * g
6. Now that we have the maximum acceleration that static friction can provide, we can calculate the time required to reach 33.0 m/s. We can use the following equation of motion:
v = u + a * t
where v is the final velocity (33.0 m/s), u is the initial velocity (0 m/s), a is the acceleration, and t is the time.
Plugging in the values, we get:
33.0 = 0 + (μs * g) * t
Solving for t:
t = 33.0 / (μs * g)
7. Finally, substitute the given values for the coefficients of static friction:
t = 33.0 / (0.750 * 9.8)
Calculating this, we find:
t ≈ 5.94 seconds
Therefore, the shortest time the truck could accelerate uniformly to 33.0 m/s without causing the box to slide is approximately 5.94 seconds.
To find the shortest time this truck could accelerate uniformly to 33.0 m/s without causing the box to slide, we need to determine the maximum acceleration that static friction can provide. Here's how you can approach this problem step-by-step:
1. Determine the maximum acceleration provided by static friction:
- Static friction is given by the equation Fs ≤ μs * N, where Fs is the magnitude of static friction, μs is the coefficient of static friction, and N is the normal force.
- In this case, the normal force is equal to the weight of the toolbox, since it is on a flat truck bed and there is no vertical acceleration.
- The weight of the toolbox is given by the equation W = mg, where m is the mass of the toolbox and g is the acceleration due to gravity.
- Substitute the weight into the normal force equation: N = W = mg.
- Maximize the static friction: Fs = μs * N = μs * mg.
- The maximum acceleration provided by static friction is equal to Fs / m: a_static = Fs / m = μs * g.
2. Calculate the maximum acceleration using the given values:
- The coefficient of static friction μs is given as 0.750.
- The acceleration due to gravity g is approximately 9.8 m/s^2.
- Substitute these values into the equation: a_static = μs * g.
3. Solve for the time required to reach 33.0 m/s:
- Use Newton's second law, F = ma, where F is the net force acting on the toolbox, m is its mass, and a is the acceleration.
- The net force is equal to the force provided by static friction: F = Fs = m * a_static.
- Rearrange the equation to solve for acceleration: a = F / m = a_static.
- We need to find the time required to reach a final velocity of 33.0 m/s, starting from rest. The initial velocity is 0.
- Use 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.
- Substitute the given values into the equation: v = u + at = 33.0 m/s, u = 0, a = a_static, and solve for t.
By following these steps, you can calculate the maximum acceleration provided by static friction and determine the shortest time required to reach the given velocity without causing the box to slide.