A crate of eggs is located in the middle of the flat bed of a pickup truck as the truck negotiates an unbanked curve in the road. The curve may be regarded as an arc of a circle of radius 36.5 m. If the coefficient of static friction between crate and truck is 0.590, how fast can the truck be moving without the crate sliding?
F(along radius) = ma
μmg=mv²/r
μg=v²/r
μgr=v²
v²=(0.590)(9.80)(36.5)
v ~= 14.5m/s
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To find the maximum speed at which the truck can be moving without the crate sliding, we need to consider the maximum static friction force that can be exerted on the crate.
The maximum static friction force (Ff) can be calculated using the following formula:
Ff = μs * N
where:
μs is the coefficient of static friction (0.590)
N is the normal force acting on the crate
The normal force (N) can be calculated as the weight of the crate (mg) minus the centripetal force (Fc):
N = mg - Fc
The weight of the crate (mg) can be calculated as the mass of the crate (m) multiplied by the acceleration due to gravity (g = 9.8 m/s^2).
Since the crate is not sliding, the centripetal force (Fc) must be equal to or less than the maximum static friction force (Ff). The centripetal force can be calculated as:
Fc = m * v^2 / r
where:
v is the velocity of the truck
r is the radius of the curve (36.5 m)
Combining the equations:
N = mg - (m * v^2 / r)
Now, substituting into the equation for Ff:
Ff = μs * (mg - (m * v^2 / r))
Since the maximum static friction force is equal to the centripetal force:
μs * (mg - (m * v^2 / r)) = m * v^2 / r
Simplifying the equation:
μs * mg - μs * (m * v^2 / r) = m * v^2 / r
μs * mg = (μs + 1) * (m * v^2 / r)
μs * g = (μs + 1) * (v^2 / r)
v^2 = (μs * g * r) / (μs + 1)
Now, we can solve for v:
v = √((μs * g * r) / (μs + 1))
Plugging in the given values:
μs = 0.590
g = 9.8 m/s^2
r = 36.5 m
v = √((0.590 * 9.8 * 36.5) / (0.590 + 1))
v ≈ √(206.615 / 1.59)
v ≈ √129.84
v ≈ 11.4 m/s
Therefore, the truck can be moving at a maximum speed of approximately 11.4 m/s without the crate sliding.
To determine the maximum speed at which the truck can be moving without the crate sliding, we can consider the forces acting on the crate.
1. Centripetal Force (Fc): The crate, moving in a circular path, requires a centripetal force to keep it in that path. The centripetal force can be calculated using the formula:
Fc = m * v² / r
Where:
- Fc is the centripetal force
- m is the mass of the crate
- v is the velocity of the truck
- r is the radius of the curve
2. Frictional Force (Ff): The static friction between the crate and the truck bed acts as a force opposing the crate from sliding. The maximum static friction force can be calculated as:
Ff = μs * N
Where:
- Ff is the frictional force
- μs is the coefficient of static friction
- N is the normal force acting on the crate (equal to the weight of the crate)
To prevent the crate from sliding, the maximum static frictional force (Ff) must be equal to or greater than the required centripetal force (Fc). Thus, we can set them equal to find the maximum velocity:
μs * N = m * v² / r
The normal force (N) is equal to the weight of the crate, which can be calculated as:
N = m * g
Where:
- g is the acceleration due to gravity (approximately 9.8 m/s²)
Combining the equations, we have:
μs * m * g = m * v² / r
Simplifying the equation, the mass of the crate cancels out, and we get:
μs * g = v² / r
Solving for v, we find:
v = √(μs * g * r)
Now we can plug in the values:
μs = 0.590
g = 9.8 m/s²
r = 36.5 m
Calculating:
v = √(0.590 * 9.8 * 36.5)
v ≈ 21.3 m/s
Therefore, the truck can be moving at a maximum speed of approximately 21.3 m/s (or about 76.7 km/h) without the crate sliding.