A light inextensible string passes over a smooth light pulley and masses of 3 and 7 kg are attached to its ends.the masses are released from rest when 7kg mass is 1m above a horizontal table. 1) show that 7kg mass hits the table about 5/9 seconds if released.
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To show that the 7 kg mass hits the table after approximately 5/9 seconds, we can use some principles of physics.
1) Determine the acceleration of the system:
Since the pulley is smooth and the string is inextensible, the tension in the string will be the same on both sides of the pulley. Let's denote this tension as T.
The net force acting on the 7 kg mass is its weight, which is given by W = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s²).
The net force is given by F = ma, where F is the net force and a is the acceleration. In this case, the net force is equal to the tension T, so we have T = ma.
Since the 7 kg mass is moving downward, we take the positive direction as downward. Therefore, the acceleration of the system is a = g - T/m.
2) Determine the relationship between the acceleration and the distance traveled:
The acceleration of an object undergoing constant acceleration is given by the equation v² = u² + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance traveled.
In this case, the initial velocity is zero because the masses are released from rest. Hence, the equation simplifies to v² = 2as.
3) Determine the distance traveled by the 7 kg mass:
The distance traveled by the 7 kg mass is the height it is above the table, which is 1 meter.
Substituting the given values into the equation, we have v² = 2as. Since the initial velocity is zero, the final velocity v is the velocity just before the 7 kg mass hits the table. Therefore, v is the same as the final velocity of the 3 kg mass, which is moving upward.
4) Find the time taken for the 7 kg mass to hit the table:
The time taken for an object to reach a certain velocity can be determined by using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken. In this case, the final velocity v is zero because the 7 kg mass hits the table and stops. The initial velocity u is the same as the final velocity of the 3 kg mass.
Substituting the given values into the equation, we have 0 = u + at.
Rearranging the equation, we have t = -u/a.
5) Calculate the time taken for the 7 kg mass to hit the table:
To calculate the time taken, we need to find the value of u and a. From step 1, we know that a = g - T/m. From step 3, we know that v² = 2as.
Using the given masses, we can find the value of T by using the equation T = m₁g - m₂g, where m₁ = 7 kg and m₂ = 3 kg.
Substituting the value of T and m into the equation a = g - T/m, we can find the value of a.
Then, substituting the value of a into v² = 2as, we can find the value of u.
Finally, substituting the values of u and a into t = -u/a, we can calculate the time taken for the 7 kg mass to hit the table.
By following these steps, you can calculate that the 7 kg mass hits the table after approximately 5/9 seconds.