A mass is pushed over a surface by a 19 N force that acts somewhat downward at an angle of θ = 45° to the horizontal as shown in the figure below. The mass is 3.0 kg and the coefficient of friction between the mass and surface is 0.25.
(a)What is the magnitude of the frictional force on the mass from the surface?
N
(b) What is the magnitude of the acceleration of the mass?
17
23
To find the magnitude of the frictional force on the mass from the surface, we need to determine two factors: the normal force and the frictional force.
1. Normal Force (N):
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, we can find the normal force by decomposing the force applied at an angle.
The vertical component of the force is given by:
Fv = F × sin(θ)
= 19 N × sin(45°)
The normal force is equal to the vertical component of the force since the mass is at rest on a horizontal surface (no vertical acceleration):
N = Fv
= 19 N × sin(45°)
2. Frictional Force (f):
The frictional force is given by:
f = μ × N
where μ is the coefficient of friction and N is the normal force we found earlier.
Now, we can substitute the values to find the magnitude of the frictional force:
f = 0.25 × N
To determine the acceleration of the mass, we can use Newton's second law of motion:
ΣF = ma
where ΣF is the sum of the forces acting on the object, m is the mass, and a is the acceleration.
The sum of the forces acting on the object are the applied force and the frictional force:
ΣF = F - f
We can substitute the known values into the equation:
ΣF = 19 N - f
Now, rearrange the equation to solve for acceleration:
ΣF = ma
19 N - f = ma
Substitute the value of f we found earlier:
19 N - (0.25 × N) = 3.0 kg × a
Simplify the equation and solve for a, the acceleration of the mass.