1.(a) A solid sphere of mass 100gm and radius 2.5cm rolls without sliding wit ha uniform velocity of 10cm/sec along a straight line on a smooth horizontal table. Calculate the total energy.
(b) A hoop of radius 100cm and mass 19kg is rolling along a horizontal surface, so that its center of mass has a velocity of 20cm/sec. How much work will have to be done to stop it?
Subject is not CUC but physics
In each case, add the translational kinetic energy, (M/2)V^2, to the rotational kinetic energy, (I/2)w^2.
I is the moment of intertia about the center of mass and w is the angular velocity.
(1a) (M/2)V^2 = (1/2)*0.050*(0.1)^2
= 2.5*10^-4 J
(I/2)w^2 = (1/2)(2/5)M*R^2*(V/R)^2
= (M/5)V^2 = 1.0*10^-4 J
Total = 3.5*10^-4 J
(1b) (M/2)V^2 = (0.5)(0.019)(0.2)^2 J
(I/2)w^2 = (0.5)M*R^2(V/R)^2
= (0.5)M*V^2
Translational and rotational KE are equalo for the hoop. The work required to stop is the negative of that.
To calculate the total energy of the rolling solid sphere (question 1a), we need to consider both its kinetic energy and rotational energy.
(a) The kinetic energy of the rolling sphere can be calculated using the equation:
KE = 0.5 * m * v^2
where m is the mass of the sphere and v is the linear velocity.
Given:
Mass of the sphere (m): 100 g = 0.1 kg
Radius of the sphere (r): 2.5 cm = 0.025 m
Linear Velocity (v): 10 cm/sec = 0.1 m/sec
First, we calculate the moment of inertia (I) for a solid sphere:
I = (2/5) * m * r^2
Next, we calculate the rotational energy (KE_rot) using the moment of inertia and angular velocity:
KE_rot = 0.5 * I * ω^2
where ω is the angular velocity.
Since the sphere is rolling without sliding, we can relate the linear velocity (v) and the angular velocity (ω) using the equation:
v = ω * r
Now, we can calculate the angular velocity (ω) based on the given linear velocity (v):
ω = v / r
Substituting this value of ω into the equation to calculate rotational energy (KE_rot), we get:
KE_rot = 0.5 * I * (v/r)^2
The total energy (E) of the rolling sphere is the sum of its kinetic energy (KE) and rotational energy (KE_rot):
E = KE + KE_rot
Substituting the calculated values, we can find the total energy.
(b) To calculate the work required to stop the rolling hoop (question 1b), we need to consider its kinetic energy.
The kinetic energy (KE) of the rolling hoop can be calculated using the equation:
KE = 0.5 * I * ω^2
where I is the moment of inertia and ω is the angular velocity.
Given:
Mass of the hoop (m): 19 kg
Radius of the hoop (r): 100 cm = 1 m
Linear Velocity (v): 20 cm/sec = 0.2 m/sec
For a hoop, the moment of inertia (I) is given by:
I = m * r^2
We can calculate the angular velocity (ω) based on the given linear velocity (v):
ω = v / r
Substituting this value of ω into the equation to calculate kinetic energy (KE), we get:
KE = 0.5 * (m * r^2) * (v/r)^2
= 0.5 * m * v^2
Therefore, the work required to stop the rolling hoop is equal to its initial kinetic energy.
Let me calculate the solutions for you.