A block is at rest on the incline shown in the
figure. The coefficients of static and kinetic
friction are µs = 0.7 and µk = 0.59, respectively.
The acceleration of gravity is 9.8 m/s^s. What is the frictional force acting on the 45 kg mass? Answer in units of N
What is the largest angle which the incline
can have so that the mass does not slide down
the incline? Answer in units of ◦.
What is the acceleration of the block down
the incline if the angle of the incline is 37 ◦? Answer in units of m/s^2.
M*g = 45*9.8 = 441 N. = Wt. of block.
Fp = 441*sin A = Force parallel with incline.
Fn = 441*Cos A = Normal force.
a. Fs = u*Fn = 0.7*441*Cos A = 308.7*Cos A = Force of static friction.
Fki = u*Fn = 0.59*441*Cos A = 260.2*Cos A = Force of kinetic friction.
b. Fp-Fs = M*a.
441*sin A-308.7*Cos A = M*0 = 0,
441*sin A/Cos A-308.7 = 0,
441*Tan A = 308.7,
A = 35 Degrees.
c. 441*sin37-260.2*Cos37 = M*a.
45a = 57.6,
a = 1.28 m/s^2.
To find the frictional force acting on the 45 kg mass, follow these steps:
Step 1: Identify the force components involved.
In this case, we have:
- The force of gravity acting straight down.
- The normal force exerted by the incline perpendicular to its surface.
- The frictional force acting parallel to the surface of the incline.
Step 2: Calculate the normal force.
The normal force can be determined using the equation:
Normal force = mass * acceleration due to gravity * cos(angle of incline)
In this case, the mass is 45 kg, and the acceleration due to gravity is 9.8 m/s^2. The angle of the incline is not given for this step, so we will leave it as cos(angle of incline).
Normal force = 45 kg * 9.8 m/s^2 * cos(angle of incline)
Step 3: Calculate the maximum static frictional force.
The maximum static frictional force can be found using the equation:
Maximum static frictional force = coefficient of static friction * normal force
In this case, the coefficient of static friction is given as µs = 0.7, and we already calculated the normal force in Step 2.
Maximum static frictional force = 0.7 * (45 kg * 9.8 m/s^2 * cos(angle of incline))
Step 4: Determine the frictional force acting on the mass.
Since the block is at rest, the frictional force acting on it will be equal to the maximum static frictional force.
Frictional force = maximum static frictional force
Now we can proceed to the next question.
To find the largest angle at which the mass does not slide down the incline, follow these steps:
Step 1: Determine the force components involved.
We have already identified the force components in the previous question.
Step 2: Equate the maximum static frictional force to the gravitational component along the incline.
Using the equation from Step 4 in the previous question:
Maximum static frictional force = mass * acceleration due to gravity * sin(angle of incline)
Step 3: Solve for the angle of incline.
Set the maximum static frictional force equal to the gravitational component along the incline and solve for the angle:
0.7 * (45 kg * 9.8 m/s^2 * cos(angle of incline)) = 45 kg * 9.8 m/s^2 * sin(angle of incline)
Simplify and solve for the angle of incline.
Now we can proceed to the final question.
To find the acceleration of the block down the incline at an angle of 37°, follow these steps:
Step 1: Determine the force components involved.
We have already identified the force components in the first question.
Step 2: Calculate the net force acting on the block.
Net force = (force of gravity component along the incline) - (force of friction)
The force of gravity component along the incline can be determined using the equation:
Force of gravity component along the incline = mass * acceleration due to gravity * sin(angle of incline)
The force of friction can be found using the equation from Step 4 in the first question:
Force of friction = 0.7 * (45 kg * 9.8 m/s^2 * cos(angle of incline))
Step 3: Calculate the acceleration of the block.
Using Newton's second law, we can calculate the acceleration:
Net force = mass * acceleration
Therefore:
(mass * acceleration) = (force of gravity component along the incline) - (force of friction)
Solve for acceleration.
To find the answers to these questions, we can apply the concepts of forces, friction, and Newton's laws of motion. Let's break down each question and the steps to find the answers.
1. What is the frictional force acting on the 45 kg mass?
We can start by calculating the gravitational force acting on the mass. The gravitational force (weight) is given by the formula: weight = mass × acceleration due to gravity. In this case, weight = 45 kg × 9.8 m/s^2 = 441 N.
The frictional force can be found using the formula: frictional force = coefficient of static friction × normal force. The normal force, in this case, is equal to the weight of the mass because the mass is at rest on the inclined plane. Therefore, the frictional force is given by: frictional force = 0.7 × 441 N = 308.7 N.
So, the frictional force acting on the 45 kg mass is 308.7 N.
2. What is the largest angle which the incline can have so that the mass does not slide down the incline?
When the object is at the verge of sliding, the force of static friction is at its maximum and preventing the object from sliding. Therefore, the maximum angle the incline can have without the mass sliding down is when the force of static friction equals the gravitational force acting downhill.
The force of static friction is given by: force of static friction = coefficient of static friction × normal force. In this case, the normal force is equal to the weight of the mass. So, force of static friction = 0.7 × 441 N = 308.7 N.
Since the force of static friction equals the gravitational force acting downhill, which is 441 N, we can write the equation as: 308.7 N = 441 N × sin(θ), where θ is the angle of the incline.
Solving for θ, we find: θ = sin^(-1)(308.7 N / 441 N) ≈ 33.12°.
Therefore, the largest angle the incline can have so that the mass does not slide down is approximately 33.12°.
3. What is the acceleration of the block down the incline if the angle of the incline is 37°?
To calculate the acceleration of the block, we need to consider the forces acting on it. The force of gravity is acting in the downward direction, and the force of static friction acts in the upward direction. The net force, which causes acceleration, is given by: net force = gravitational force − force of static friction.
The gravitational force acting downhill is given by: gravitational force downhill = weight × sin(θ). In this case, θ = 37°, so gravitational force downhill = 441 N × sin(37°) ≈ 265.4 N.
The force of static friction is given by: force of static friction = coefficient of static friction × normal force. The normal force is equal to the weight of the mass, which is 441 N. So, force of static friction = 0.7 × 441 N = 308.7 N.
The net force then becomes: net force = gravitational force downhill − force of static friction = 265.4 N − 308.7 N = -43.3 N. The negative sign indicates that the net force opposes the motion down the incline.
Using Newton's second law of motion, net force = mass × acceleration, we can solve for the acceleration: -43.3 N = 45 kg × acceleration.
Solving for acceleration, we find: acceleration = -43.3 N / 45 kg ≈ -0.96 m/s^2.
Therefore, the acceleration of the block down the incline, when the angle of the incline is 37°, is approximately -0.96 m/s^2. The negative sign indicates that the block is moving uphill.