1.A net force that is linearly increasing in time is applied to an object. What is time dependence of the object's kinetic energy?
2.A tire is rolling along a road, without slipping, with a center of mass velocity of v. A piece of tape is attached to the tire. When the tape is at the top of the tire, what is its velocity with respect to the road?
1. d(KE)/dt = (1/2)M dV^2/dt = M V dV/dt
acceleration = dV/dt = a t where a is a constant, since force increases linearly with time
V = a t^2/2 assuming V=0 at t=0
d(KE)/dt = a M t^2/2 * at
= (1/2) m a^2 t^3
2. 2 v.
The point of contact with the road is the instantaneous center of motion. The top of the tire is twice the distance from the road contact point, compared to the axle.
t^3
1. To determine the time dependence of the object's kinetic energy when a net force that is linearly increasing with time is applied, we need to consider the relationship between force, velocity, and kinetic energy.
The net force applied to an object is given by the equation F = m * a, where F is the force, m is the mass of the object, and a is the acceleration of the object.
The acceleration of an object is related to its velocity by the equation a = dv/dt, where v is the velocity of the object and t is time.
The work done on an object is the force applied multiplied by the displacement, and is given by the equation W = F * d, where W is the work done, F is the force, and d is the displacement.
The kinetic energy of an object is given by the equation KE = 1/2 * m * v^2, where KE is the kinetic energy, m is the mass of the object, and v is the velocity of the object.
When a net force is applied to an object, the work done on the object is equal to the change in kinetic energy of the object. Therefore, we can say that W = ΔKE, where ΔKE is the change in kinetic energy.
In this case, since the net force is linearly increasing with time, we can say that the force can be represented as F = k * t, where k is a constant.
By substituting this into the equation for work done, we have W = (k * t) * d.
Since the net force is doing work on the object, it is increasing the kinetic energy of the object. Therefore, the change in kinetic energy can be represented as ΔKE = (k * t) * d.
To find the time dependence of the object's kinetic energy, we need to differentiate the above equation with respect to time. This will give us the rate of change of kinetic energy with respect to time, also known as the time derivative of kinetic energy.
Therefore, d(ΔKE)/dt = k * d.
From this, we can conclude that the time dependence of the object's kinetic energy when a net force that is linearly increasing with time is applied is given by d(ΔKE)/dt = k, where k is a constant.
2. When the tape is at the top of the tire, its velocity with respect to the road can be determined using the concept of relative velocities.
Let's assume that the tire is moving to the right with a velocity v, and the tape is attached to a point on the circumference of the tire at the top, moving along with the tire without slipping.
Since the tape is attached to the tire, it inherits the same velocity as the tire at any given instant. Therefore, at the top of the tire, the tape has the same velocity as the tire, which is v.
Hence, the velocity of the tape with respect to the road when it is at the top of the tire is also v.