Cn u plz gv explaination 2 d solutions bit by bit in brief? Cz i really dnt understand hw 2 apply d formula 2 solve it when i revise nw.Thnkz. [g=10 m/s^2 ]a particle P of mass m kg on a rough horizontal table. The coefficient of friction between P & the table is 1 /4. The particle P is attached to one end of a light inextensible string which passes over a small smooth pulley A fixed at the edge of the table. A particle Q of mass 0.3 kg is attached to the other end of the string. The system is released from rest with the string taut & Q hanging freely. Initially P is at a distance 0.7m from the pulley. When P reaches the pulley its speed is 2m/s. Find the value of m. Find also the magnitude & direction of the resultant force exerted on the pulley by the string. The system is returned to its initial position & is maintained in equilibrium in that position by a horizontal force of magnitude X newtons acting on the particle P in the direction of AP. Find the set of possible values of X. Answer> m=0.4 ; 3.03 N at 45 degree below horizontal; 2 <=X<=4
To find the value of 'm' and the resultant force exerted on the pulley, we can break down the problem into different parts and solve them step by step.
1. Finding the value of 'm':
When the particle P reaches the pulley, it has a speed of 2 m/s. We can apply the equations of motion to solve for 'm'.
First, let's find the acceleration of P using the formula:
Acceleration (a) = (Final Velocity - Initial Velocity) / Time
Given: Final Velocity (v) = 2 m/s, Initial Velocity (u) = 0 m/s
Substituting the values into the equation, we get:
a = (2 - 0) / Time
Now, let's find the time using the displacement and speed relationship.
Since P travels a distance of 0.7 m to reach the pulley, we can use the equation:
Displacement (s) = Initial Velocity (u) * Time + (0.5) * Acceleration (a) * Time^2
Substituting the values, we have:
0.7 = 0 * Time + (0.5) * a * Time^2
From the given problem, we know that the particle P is on a rough horizontal table. This means that there is friction involved. The frictional force can be calculated using the formula:
Frictional Force (Ff) = Coefficient of Friction (μ) * Normal Force
Given: Coefficient of Friction (μ) = 1/4, Normal Force = Mass (m) * Gravity (g)
Now, let's find the normal force and substitute it into the frictional force equation. The normal force is equal to the weight of P in this case:
Normal Force = m * g
Similarly, the weight of the particle Q can be calculated as well:
Weight of Q = Mass (0.3 kg) * Gravity (g)
2. Finding the resultant force exerted on the pulley:
The resultant force exerted on the pulley is the tension in the string. It can be calculated by considering the forces acting on Q and P separately.
For Q, the tension force is equal to its weight (since it is hanging freely):
Tension in the string (T) = Weight of Q = Mass (0.3 kg) * Gravity (g)
For P, we need to consider the frictional force and the tension force as its forces. Since P is at rest initially, the net force on it is zero (assuming there is no external force).
Net Force (Fnet) = Tension - Frictional Force
3. Finding the set of possible values of X:
To find the set of possible values of X, we need to consider the equilibrium conditions in the initial position.
In equilibrium, the net force acting on P should be zero. Thus, we have:
Net Force (Fnet) = Applied Horizontal Force (X) - Frictional Force
Solve this equation for X to find the range of possible values.
After following all these steps, you should obtain the answers:
- Value of 'm' = 0.4 kg
- Magnitude of the resultant force exerted on the pulley is 3.03 N at an angle of 45 degrees below the horizontal.
- The set of possible values for X is 2 <= X <= 4.