At a carnival, you can try to ring a bell by striking a target with a 11.6-kg hammer. In response, a 0.413-kg metal piece is sent upward toward the bell, which is 4.22 m above. Suppose that 26.2 percent of the hammer's kinetic energy is used to do the work of sending the metal piece upward. How fast must the hammer be moving when it strikes the target so that the bell just barely rings?
0.262KE=PE
0.262m₁v²/2 = m₂gh
v=sqrt{2m₂gh/0.262m)=
=sqrt{2•0.413•9.8•4.22/0.262•11.6} =
=3.35 m/s
To find the speed at which the hammer must be moving to just barely ring the bell, we can use the principle of conservation of energy. The kinetic energy of the hammer is transferred to the metal piece, which then does work against gravity to raise the metal piece.
Let's break down the problem step by step:
Step 1: Find the initial kinetic energy of the hammer.
The kinetic energy (KE) is given by the equation KE = (1/2)mv^2, where m is the mass and v is the velocity of an object.
Given:
Mass of the hammer, m_hammer = 11.6 kg
We need to find the velocity of the hammer, v_hammer, when it strikes the target so that the bell just barely rings. This is the initial velocity of the hammer.
Step 2: Find the kinetic energy transferred to the metal piece.
Given:
Percent of kinetic energy transferred to the metal piece, p = 26.2%
The kinetic energy transferred to the metal piece is a fraction of the hammer's initial kinetic energy. Let's calculate it:
Initial kinetic energy of the hammer, KE_hammer = (1/2) m_hammer * v_hammer^2
Kinetic energy transferred to the metal piece, KE_transfer = p * KE_hammer
Step 3: Use the conservation of energy to find the velocity of the metal piece when it reaches the bell.
The kinetic energy transferred to the metal piece is converted into potential energy when it reaches the height of the bell, which is then equal to its gravitational potential energy.
Gravitational potential energy (PE_gravity) is given by the equation PE_gravity = m_piece * g * h, where m_piece is the mass of the metal piece, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height (4.22 m).
Equating KE_transfer to PE_gravity, we have:
KE_transfer = m_piece * g * h
Rearranging the equation to solve for the velocity of the metal piece, v_piece:
v_piece = sqrt(2 * KE_transfer / m_piece)
Step 4: Solve for the initial velocity of the hammer.
We know that the velocity of the hammer when it strikes the target is equal to the velocity of the metal piece at the start. Therefore, v_hammer = v_piece.
By substituting this value into the equation for v_piece, we can solve for v_hammer:
v_hammer = sqrt(2 * KE_transfer / m_piece)
Putting it all together, we can find the velocity of the hammer when it strikes the target so that the bell just barely rings.