A crate of mass 43 kg is loaded onto the back of a flatbed truck. The coefficient of static friction between the box and the truck bed is 0.47. What is the maximum acceleration a (in m/s2) the truck can have without the crate sliding off?
Draw a picture and label the forces to better understand the question.
Sum the forces in the x direction since its asking for acceleration (motion is usually in x direction so you're being asked to find a_x (x- axis).
Sum of F_x = ma_x
The only force on x axis is static friction (mu_s) pointing toward the box in a direction that will keep it from sliding as the truck moves.
so,
mu_s = ma_x
we know that mu*N = static friction
so solve for normal force by summing forces on Y axis = N - mg =0 (no acceleration on Y axis)
N = mg
Finally,
mu_S (mg) = ma_x
solve for a_x. notice that masses cancel
thus:
mu_S*g = a_x
Final answer: 4.61
Hope that wasn't too confusing. If you can draw/label the diagram properly you can solve it properly too! :)
To determine the maximum acceleration (a) the truck can have without the crate sliding off, we need to calculate the maximum static friction force that can be exerted between the crate and the truck bed.
The maximum static friction force can be found using the formula:
F_max = μ_s * N,
where F_max is the maximum static friction force, μ_s is the coefficient of static friction, and N is the normal force.
The normal force is equal to the weight of the crate, which can be calculated using the formula:
N = m * g,
where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values into the formula:
N = 43 kg * 9.8 m/s^2,
N = 421.4 N.
Now, we can calculate the maximum static friction force:
F_max = 0.47 * 421.4 N,
F_max ≈ 197.9 N.
Finally, we can calculate the maximum acceleration:
F_max = m * a,
197.9 N = 43 kg * a,
a ≈ 4.6 m/s^2.
Therefore, the maximum acceleration the truck can have without the crate sliding off is approximately 4.6 m/s^2.
To find the maximum acceleration the truck can have without the crate sliding off, we need to consider the forces acting on the crate. There are two main forces to consider: the gravitational force (mg) pulling the crate downward, and the static friction force (fs) between the crate and the truck bed, which acts in the opposite direction of motion and prevents the crate from sliding.
The maximum static friction force can be found using the formula: fs = μs * N
where μs is the coefficient of static friction and N is the normal force.
The normal force (N) can be calculated as the force exerted by the truck bed on the crate, and it is equal to the weight of the crate (mg) since the crate is at rest on a horizontal surface.
Therefore, N = mg = (mass of the crate) * (acceleration due to gravity) = 43 kg * 9.8 m/s^2.
Using the coefficient of static friction (μs = 0.47) and the normal force (N = 43 kg * 9.8 m/s^2), we can calculate the maximum static friction force:
fs = μs * N = 0.47 * (43 kg * 9.8 m/s^2).
Now, the maximum acceleration can be found by equating the maximum static friction force to the force that would cause the crate to slide off. In this case, it would be the force due to acceleration.
Therefore, fs = ma
Substituting the value of fs, we get:
0.47 * (43 kg * 9.8 m/s^2) = 43 kg * a
Simplifying the equation, we find:
a = (0.47 * 43 kg * 9.8 m/s^2) / (43 kg)
a = 0.47 * 9.8 m/s^2
Calculating this, we get:
a ≈ 4.61 m/s^2
So, the maximum acceleration the truck can have without the crate sliding off is approximately 4.61 m/s^2.