Can you explain the solutions steps by steps? Because i really don't understand when i revise it now. Thanks. The lines CA & CB are lines of greatest slope of two smooth fixed planes that intersect in a horizontal line through C. AB is horizontal & 5
m long. BC=3m & CA=4m. Particles P & Q, having masses 0.3kg & 0.4kg respectively, are placed on CA & CB, & are joined by a light inextensible string that passes over a small pulley at C. The particles are held at rest with the string taut & are then released. Find the magnitude of the acceleration of each particle. During the motion a force of magnitude X N directed towards A is applied to P. As a result, the acceleration of P is reversed in direction but is unchanged in magnitude. Find X. Given that the force of magnitude X N is applied 0.25s after the particles are released, find the distance travelled by P before it first comes instantaneously to rest. [you may assume that P does not reach C & Q does not reach B.] answers: 1.96ms^-1 ; 2.75N ; 0.061m
To solve this problem, we can break it down into several parts. Let's go step by step:
1. First, let's find the magnitude of acceleration for each particle:
We know that the mass of particle P is 0.3kg and the mass of particle Q is 0.4kg.
Applying Newton's second law, F = ma, we can find the acceleration for each particle.
We have two sets of forces acting on each particle: the weight force and the tension force in the string.
The weight force acting on P is given by Wp = mp * g, where mp is the mass of P and g is the acceleration due to gravity (approximated as 9.8 m/s^2).
Similarly, the weight force acting on Q is given by Wq = mq * g, where mq is the mass of Q.
The tension in the string is the same for both particles and can be denoted as T.
Now, considering the forces acting on P in the vertical direction, we have:
T - Wp = mp * ap [ap represents the acceleration of particle P]
And for Q, we have:
T - Wq = mq * aq [aq represents the acceleration of particle Q]
Substituting Wp = mp * g and Wq = mq * g, we can rewrite the above equations as:
T - mp * g = mp * ap [1]
T - mq * g = mq * aq [2]
We also know that ap = -aq (reversed in direction but same magnitude). This means that the acceleration of P is negative and equal in magnitude to the acceleration of Q.
By solving equations [1] and [2] simultaneously, we can find the values of T, ap, and aq.
2. Next, let's find the magnitude of force X:
We are given that the acceleration of P is reversed in direction but unchanged in magnitude when force X is applied.
So, the net force acting on P after force X is applied is:
F_net = T - mp * g + X
Since the acceleration is unchanged, we can equate the net force to mp * ap:
mp * ap = T - mp * g + X [3]
We can solve equation [3] to find the value of X.
3. Finally, let's find the distance traveled by P before it first comes instantaneously to rest:
To find this distance, we can use the equation of motion:
v^2 = u^2 + 2as
where v is the final velocity (which is 0 since P comes to rest), u is the initial velocity, a is the acceleration, and s is the distance traveled.
We know the final velocity is 0, the initial velocity is 0 (since P is at rest initially), and we need to find the distance s. We already know the acceleration from step 1.
By solving the equation above for s, we can find the distance traveled by P before it comes to rest.
Now that we have the step-by-step plan, let's proceed with the calculations using the given values:
1. Find the magnitude of acceleration for each particle:
Given:
Mass of P, mp = 0.3kg
Mass of Q, mq = 0.4kg
Acceleration due to gravity, g = 9.8 m/s^2
Using equations [1] and [2], we can solve for T, ap, and aq.
2. Find the magnitude of force X:
Given:
Acceleration of P, ap = -aq (reversed in direction)
Using equation [3], we can solve for X.
3. Find the distance traveled by P before it first comes instantaneously to rest:
Given:
Initial velocity, u = 0
Final velocity, v = 0
Acceleration, a = ap (from step 1)
Using the equation of motion, we can solve for s to find the distance traveled by P.
By following these steps, you should be able to find the values for the magnitude of acceleration for each particle, the magnitude of force X, and the distance traveled by P.