A force sensor is used to measure a force applied to a 9.1 kg box on a hard, horizontal surface. It records that the box starts to move when the force is equal to 26.9 N. This force makes the box start to move with an acceleration of 0.2 m/s2. Find mu sub s/k respectively.
F(fr) =μ•N = μ•m•g
1. F(fr)=F
μ(s)•m•g =F,
μ(s) =F/m•g =26.9/9.1•9.8 = 0.3
2. m•a=F(fr)
m•a = μ(k)•m•g
μ(k)=a/g=0.2/9.8=0.2
First compute mu(static):
When the box just begins to move, the applied force equals mu(s)*m*g
So mu(s)= F/m*g = 26.9/(9.1*9.8)= 0.30
Now when the box gets into motion, mu(k) comes in picture and frictional force f = mu(k)*m*g
So F - mu(k)*m*g = m*a
mu(k) = (F - m*a)/m*g
= (26.9-9.1*0.2)/9.1*9.8
= 0.28
To find the coefficient of static friction (μs) and the coefficient of kinetic friction (μk), we need to use the given information about the force and acceleration. Here's how you can calculate them step by step:
1. Calculate the force of gravity acting on the box:
The force of gravity (weight) is calculated using the formula:
Fg = m * g
where m is the mass of the box (9.1 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Fg = 9.1 kg * 9.8 m/s^2 = 89.18 N
2. Calculate the force of static friction (fs):
The force of static friction can be calculated using the formula:
fs = μs * Fn
where μs is the coefficient of static friction and Fn is the normal force acting on the box.
Since the box is on a horizontal surface, the normal force (Fn) is equal to the force of gravity (Fg). So, Fn = Fg = 89.18 N.
Therefore, fs = μs * 89.18 N.
3. Determine the minimum force required to overcome static friction:
The minimum force required to overcome static friction and start the box moving is equal to the force of static friction:
fs = 26.9 N (given in the problem).
4. Calculate μs:
We can now equate the force of static friction to the given force to find μs:
fs = μs * Fn
26.9 N = μs * 89.18 N
μs = 26.9 N / 89.18 N = 0.301.
Therefore, the coefficient of static friction (μs) is approximately 0.301.
5. Calculate the force of kinetic friction (fk):
Once the box starts to move, the force of kinetic friction is calculated using the formula:
fk = μk * Fn
where μk is the coefficient of kinetic friction.
Given that the box starts to move with an acceleration of 0.2 m/s^2, we can calculate the net force acting on the box:
Net force (Fnet) = m * a
Fnet = 9.1 kg * 0.2 m/s^2 = 1.82 N
Since the force of kinetic friction (fk) acts opposite to the direction of motion, it is equal in magnitude to the net force:
fk = Fnet = 1.82 N
6. Calculate μk:
Using the formula for the force of kinetic friction, we can find μk:
fk = μk * Fn
1.82 N = μk * 89.18 N
μk = 1.82 N / 89.18 N = 0.0204.
Therefore, the coefficient of kinetic friction (μk) is approximately 0.0204.
To summarize:
Coefficient of static friction (μs) ≈ 0.301
Coefficient of kinetic friction (μk) ≈ 0.0204.