A box rests on the back of a truck. The coefficient of static friction between the box and the bed of the truck is 0.300. (a) When the truck accelerates forward, what force accelerates the box? (b) Find the maximum acceleration the truck can have before the box slides.
Razan is moving to a new apartment and puts a dresser in the back of her pickup truck. When the truck accelerates forward, what force accelerates the dresser? Under what circumstances could the dresser slide? In which direction?
(a) Well, let's analyze the situation here. When the truck accelerates forward, there are a few forces acting on the box. The force of gravity is pulling it downward, and the normal force from the truck's bed is pushing it upward. However, the force that actually accelerates the box forward is the force of friction. So, the force of static friction between the box and the bed of the truck accelerates the box forward.
(b) To find the maximum acceleration the truck can have before the box slides, we need to consider the maximum static friction force. The maximum static friction force is equal to the coefficient of static friction multiplied by the normal force.
Let's say the mass of the box is m, and the acceleration of the truck is a. The normal force acting on the box is equal to its weight, which is m times the acceleration due to gravity, g.
Now, the maximum static friction force is given by:
F(static friction) = coefficient of static friction * normal force
F(static friction) = coefficient of static friction * m * g
For the box to slide, the force of static friction must be equal to or greater than the force required to overcome its inertia, which is m * a.
Therefore, the maximum static friction force is equal to m * a. Setting this equal to the equation we found earlier:
m * a = coefficient of static friction * m * g
Now, we can cancel out the mass of the box, m:
a = coefficient of static friction * g
So, the maximum acceleration the truck can have before the box slides is equal to the coefficient of static friction multiplied by the acceleration due to gravity.
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Why don't scientists trust atoms?
Because they make up everything!
To answer both parts of the question, we need to understand the concept of static friction and its relationship with acceleration.
(a) When the truck accelerates forward, the force that accelerates the box is the force of static friction between the box and the bed of the truck. According to Newton's second law of motion (F = ma), the net force acting on the box is equal to the product of its mass (m) and acceleration (a). In this case, the net force comes from the force of static friction.
However, the force of static friction can vary depending on the applied force and the coefficient of static friction (μs). The maximum force of static friction can be calculated using the formula:
Fmax = μs * N
Where Fmax is the maximum force of static friction and N is the normal force exerted on the box by the truck's bed. In this case, since the box rests on the back of the truck, the normal force exerted on the box is equal to its weight (mg), where g is the acceleration due to gravity.
Therefore, the maximum force of static friction (Fmax) can be calculated as:
Fmax = μs * mg
(b) To find the maximum acceleration the truck can have before the box slides, we can equate the force of static friction with the maximum force of static friction (Fmax). By setting these two forces equal to each other, we can solve for acceleration (a).
Fs = Fmax
ma = μs * mg
Canceling out the mass (m) on both sides:
a = μs * g
Substituting the given coefficient of static friction (μs = 0.300) and the acceleration due to gravity (g = 9.8 m/s^2), we can calculate the maximum acceleration (a) the truck can have before the box slides.
a = 0.300 * 9.8
a ≈ 2.94 m/s^2
Therefore, the maximum acceleration the truck can have before the box slides is approximately 2.94 m/s^2.
(a) The friction force accelerates the box.
(b) The box slides if mus*(M g) < M a
mus = 0.30 < a/g
"mus" is the static friction coefficient. It would normally be written as a Greek lower case "mu" with an "s" subscript.
a = 0.3 g = 2.94 m/s^2 is the maximum acceleration without slipping