A small block of mass m1 = 0.500 kg is released from rest at the top of acurved-shaped frictionless wedge of mass m2 =3.00 kg, which sits on a frictionless horizontal surface as inFigure P9.60a. When the block leaves the wedge, its velocity ismeasured to be v1 = 3.20 m/s to the right, as in Figure P9.60b.
To solve this problem, we can use the principle of conservation of mechanical energy. Here's how:
1. The first step is to determine the initial potential energy of the block when it is at the top of the wedge. The formula for gravitational potential energy is given by PE = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the height of the block above some reference point. In this case, the reference point is chosen to be the bottom of the wedge. The height at the top of the wedge is the sum of the heights of the block and the wedge. Let's call this height "H." Therefore, the potential energy of the block at the top of the wedge is given by PE_block = m1 * g * H.
2. The next step is to determine the final kinetic energy of the block when it leaves the wedge. The formula for kinetic energy is given by KE = 0.5 * m * v^2, where m is the mass of the object, and v is the velocity of the object. In this case, the mass of the block is m1, and the velocity is v1. Therefore, the kinetic energy of the block when it leaves the wedge is given by KE_block = 0.5 * m1 * v1^2.
3. The block loses some potential energy as it falls down the curve of the wedge due to the conversion to kinetic energy. The change in potential energy is equal to the change in kinetic energy. Therefore, the change in potential energy is given by ΔPE = PE_block - KE_block.
4. Since the wedge is frictionless, no external work is done on the system. Therefore, the change in mechanical energy is equal to zero. ΔE = ΔPE = 0. This means that the change in potential energy is equal to the change in kinetic energy within the system.
5. Solving for ΔPE, we get ΔPE = ΔKE, which can be rewritten as m1 * g * H = 0.5 * m1 * v1^2.
6. We can now solve for the height of the wedge (H) using the given values of m1 (0.500 kg) and v1 (3.20 m/s). Plugging in these values, we get 0.500 kg * 9.8 m/s^2 * H = 0.5 * 0.500 kg * (3.20 m/s)^2. Solving for H, we find H = (0.5 * (3.20 m/s)^2) / (0.500 kg * 9.8 m/s^2).
By following these steps, we can calculate the height of the wedge (H).