Two block are connected by a rope that runs over a pulley. The block on the tables has mass 4kg, the hanging block has mass 2kg, and the pulley has mass 0.5kg and radius 0.25m. Assume that the table is friction-less. If the block are released from the rest, determine their speeds after the hanging block has dropped 0.75m.
Please explain the formula and steps. Thanks.
m1 =4 kg, m2 = 2 kg, m = 0.5 kg, R = 0.25 m, h= 0.75 m.
Projections of the equation according to the 2 Newton's law for two blocks on the horizontal (for the 1st block)and on the vertical (fot the 2nd block) axis:
m1•a = T1
m2•a =m2•g-T2,
The equation of the pulley motion (2nd Nerton's law, for the rotational motion)
I•ε =M.
The moment of inertia of the pulley (disk) is
I =m•R²/2 ,
M = torque = (T1-T2)•R,
ε = a/R,
I•ε =M => m•R²•a/2•R =(T1-T2) •R =>
m•a/2 = (T1-T2).
m1•a + m2•a = T1 + m2•g -T2 = m2•g + (T1-T2) = m2•g +m•a/2,
a = m2•a/[m1+m2-m(m/2)] =
= 2•9.8/(4+2+0.125)=3.336 m/s^2,
a = v^2/2•h ,
v=sqrt(2•a•h) = sqrt(2•3.336•0.75) =
= 2.2 m/s^2
To determine the speeds of the blocks after the hanging block has dropped 0.75m, we can use the concept of conservation of energy.
Step 1: Calculate the potential energy of the hanging block when it has dropped 0.75m.
- The potential energy of an object is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
- In this case, m = 2kg, g = 9.8m/s^2, and h = 0.75m.
- Substitute the values into the formula: PE = 2kg × 9.8m/s^2 × 0.75m = 14.7kg × m^2/s^2.
Step 2: Calculate the distance that the hanging block has traveled.
- The distance traveled by the hanging block is equal to the distance traveled by a point on the circumference of the pulley.
- Since the rope is wrapped around the pulley and the hanging block has dropped 0.75m, the distance traveled by the hanging block is equal to the change in arc length of the pulley.
- The arc length of a circle is given by the formula s = rθ, where s is the arc length, r is the radius, and θ is the angle (in radians) subtended by the arc.
- In this case, the hanging block has dropped 0.75m, which means the change in arc length is equal to the circumference of the pulley.
- The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius.
- Substitute the values to calculate the distance traveled: s = 2π × 0.25m = 1.57m.
Step 3: Calculate the kinetic energy of the two blocks after the hanging block has dropped 0.75m.
- The total kinetic energy of the system is equal to the sum of the kinetic energies of the individual blocks.
- The kinetic energy of an object is given by the formula KE = (1/2)mv^2, where m is the mass and v is the velocity.
- We can assume that the velocity of the two blocks is the same since they are connected by a rope and move together.
- Let's denote the common velocity of the two blocks as v.
For the hanging block:
- The kinetic energy of the hanging block is given by the formula KE = (1/2)mv^2.
- Substitute the values into the formula: KE1 = (1/2)(2kg)(v^2).
For the block on the table:
- The kinetic energy of the block on the table is given by the formula KE = (1/2)mv^2.
- Substitute the values into the formula: KE2 = (1/2)(4kg)(v^2).
Step 4: Apply the principle of conservation of energy.
- The principle of conservation of energy states that the total energy of a closed system remains constant.
- In this case, the initial potential energy of the hanging block is converted into the kinetic energy of the two blocks after the hanging block has dropped 0.75m.
- Therefore, we can equate the initial potential energy to the final kinetic energy: PE = KE1 + KE2.
- Substitute the values into the equation: 14.7kg × m^2/s^2 = (1/2)(2kg)(v^2) + (1/2)(4kg)(v^2).
Step 5: Solve the equation to find the velocity (speed) of the two blocks.
- Simplify the equation: 14.7 = v^2 + 2v^2.
- Combine like terms: 14.7 = 3v^2.
- Divide both sides by 3: v^2 = 4.9.
- Take the square root of both sides: v = ±2.2m/s.
Since speed cannot be negative, the final speeds of the two blocks after the hanging block has dropped 0.75m are 2.2m/s.