An inclined plane makes an angle of 30.0◦ above the horizontal and is 4.00 m long measured along the slope. A block is placed at the top of the plane and released from rest. Calculate the speed of the block when the block reaches the bottom of the plane. Assume the plane is smooth so that no friction force acts on the block.
the height (h) of the top of the plane is ... 4.00 m * sin(30.0º)
velocity (v) at the bottom is ... √(2 g h)
To solve this problem, we can use the principle of conservation of energy. Since the plane is frictionless, the mechanical energy of the block will be conserved throughout its motion.
The mechanical energy of the block consists of two components: potential energy and kinetic energy. At the top of the plane, the block only has potential energy, and at the bottom of the plane, it only has kinetic energy.
1. First, let's find the potential energy of the block at the top of the plane. The potential energy (PE) is given by the formula:
PE = m * g * h
where m is the mass of the block, g is the acceleration due to gravity, and h is the vertical height of the plane. In this case, h can be found by multiplying the length of the plane (4.00 m) by the sine of the angle (30 degrees):
h = 4.00 m * sin(30 degrees)
2. Next, let's find the kinetic energy of the block at the bottom of the plane. The kinetic energy (KE) is given by the formula:
KE = (1/2) * m * v^2
where v is the velocity of the block. At the bottom of the plane, the potential energy is zero, so all the energy is converted into kinetic energy:
KE = PE = m * g * h
3. Now, we can equate the potential energy at the top to the kinetic energy at the bottom to find the velocity. Rearranging the equation, we have:
(1/2) * m * v^2 = m * g * h
Dividing both sides by m and canceling out the m on both sides, we get:
(1/2) * v^2 = g * h
Multiplying both sides by 2, we have:
v^2 = 2 * g * h
Taking the square root of both sides, we finally get:
v = sqrt(2 * g * h)
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the vertical height of the plane obtained in step 1.
4. Plugging in the values, calculate the speed of the block when it reaches the bottom of the plane.