A bock of mass 3kg starts from rest and slides down a surface , which corresponds to a quarter of a circle of 1.6m radius.
a) if the curved surface is smooth , calculate the speed of the of the block at the bottom.
If the quarter circle has the radii horizontal and vertical, the difference in height equals the radius=1.6m.
Neglecting friction and other dissipative forces, then conservation of energy holds, and so
KE+PE=0,or
(1/2)mv^2 + mg(h-h0)=0
Solve for v
where h=0, h0=1.6m
and g=9.81 m/s²
To find the speed of the block at the bottom of the quarter-circle, we can use conservation of energy.
1) Calculate the potential energy at the starting point:
Potential energy = mass * gravity * height
Since the block starts from rest, its height at the starting point is the radius of the quarter-circle, which is 1.6m.
Potential energy at the starting point = 3kg * 9.8 m/s^2 * 1.6m = 47.04 J
2) Calculate the kinetic energy at the bottom:
When the block reaches the bottom of the quarter-circle, all of its potential energy is converted to kinetic energy.
Kinetic energy = 1/2 * mass * velocity^2
Since the block starts from rest, its initial kinetic energy is zero.
So, kinetic energy at the bottom = Potential energy at the starting point = 47.04 J
3) Solve for velocity using the kinetic energy equation:
47.04 J = 1/2 * 3kg * velocity^2
Divide both sides by 1.5kg:
31.36 m^2/s^2 = velocity^2
Taking the square root of both sides:
velocity = √31.36 m/s or approximately 5.60 m/s
Therefore, the speed of the block at the bottom of the quarter-circle is approximately 5.60 m/s.
To calculate the speed of the block at the bottom, we can use the principles of conservation of energy.
First, let's find out the potential energy at the top and the kinetic energy at the bottom.
Potential energy at the top: The block is at the top of the curve, so its height from the ground is the radius of the circle, which is 1.6m.
The potential energy is given by the formula: PE = m * g * h, where m is the mass (3kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height (1.6m).
PE = 3kg * 9.8 m/s² * 1.6m = 47.04 Joules
Kinetic energy at the bottom: At the bottom of the curve, the block has no height (h = 0), so all of its potential energy has been converted to kinetic energy.
The kinetic energy is given by the formula: KE = 1/2 * m * v², where m is the mass (3kg) and v is the velocity at the bottom (what we want to find).
KE = 47.04 Joules
Equating the potential and kinetic energy:
PE = KE
47.04 J = 1/2 * 3kg * v²
47.04 J = 1.5kg * v²
v² = 47.04 J / 1.5kg
v² = 31.36 m²/s²
To find the velocity, we take the square root of both sides:
v = √(31.36 m²/s²)
v ≈ 5.6 m/s
Therefore, the speed of the block at the bottom is approximately 5.6 m/s.