1.A block of masses mA is on a plane inclined at an angle θ with the horizontal. It is attached to another mass mB by means of string that passes over a pulley at the top of the incline. For mA = 8kg, mB = 5kg, θ = 20o, and the coefficient of kinetic friction between mA and the plane is 0.3. Calculate the (a) acceleration of the masses and (b) the tension in the string when the system is moving.
It would help with visualising if you drew block diagrams for the question.
Net force on B = downwards weight (mg) and upwards Tension (T)
Net force on A = down the incline weight (mgsinθ) and up the incline Tension (T)
I'm gonna take mass of A as M, and mass of B as m. The accelerations must be same, and are thus 'a' in both cases:
The equations formed are as follows:
1) T - Mgsinθ = Ma
2) mg - T = ma
Adding the two and eliminating T:
mg - Mg(sin20) = (M+m)a
Putting in the values:
5*10 - 8*10*0.34 = (8+5)a
=> a = (50-27.2)/13
= 22.8/13
= 1.75 m/sec^2
Now, for the Tension, you can simply put the value of a into one of the earlier equations, 1 or 2.
What happens to the coefficient of friction? Why is it not used?
To calculate the acceleration of the masses and the tension in the string, we can use Newton's second law and the concept of friction.
(a) Calculate the acceleration of the masses:
1. Begin by drawing a free-body diagram for each mass to consider all the forces acting on them.
- For mass mA, the forces acting on it are its weight (mg), the normal force (N), friction force (f), and the tension in the string (T).
- For mass mB, the forces acting on it are its weight (mg) and the tension in the string (T).
2. Break down the weight of each mass into its components parallel and perpendicular to the inclined plane.
- For mass mA, the component of weight parallel to the inclined plane is mAg*sin(θ), and the component perpendicular to the inclined plane is mAg*cos(θ).
- For mass mB, the weight is entirely acting vertically downward, so there are no components to consider in this direction.
3. Apply Newton's second law (F = ma) to each object along the x and y directions separately.
- For mass mA, in the x-direction: T - f = mA*a (equation 1).
- For mass mA, in the y-direction: N - mAg*cos(θ) = 0 (equation 2).
- For mass mB, in the y-direction: T - mBg - mB*g = mB*a (equation 3).
4. Solve the system of equations formed by equations 1, 2, and 3 to find the acceleration (a).
(b) Calculate the tension in the string:
To determine the tension in the string, we can use equation 3 from the previous step, since it includes the tension.
1. Substitute the value of the calculated acceleration (a) into equation 3.
2. Solve the resulting equation to find the tension in the string.
By following the steps outlined above, you will be able to calculate both the acceleration of the masses and the tension in the string for the given values of mass and angle.